Course: Mathematics, Topic: Unit1:Reading, Writing and Comparing Whole Numbers Beyond 1,000,000

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Unit1:Reading, Writing and Comparing Whole Numbers Beyond 1,000,000
- MATH P6 Folder
- 1) What is the place value of 5 in the number 4,565,270? Quiz
- 2) Write the numbers from the greatest to the smallest 1,673,421; 1,065,345; 1,671,241; 1,065,234 Quiz
- 3) Workout 602,444 + 2,622,433 + 5,789,987 = Quiz
- Scripted lesson Math p6 URL
- Key unit competence: To be able to round off decimals, convert fractions to decimals and vice versa.
  
  Introduction
  In counting and mathematical operations, some numbers may be found difficult to be memorized or used. These can be simplified to their nearest numbers which can be easily memorized or used. For example money used by different schools per year can be rounded to simple figures to summarize.
  Let’s consider an example of school party where Primary 6 learners contributed to buy all items needed. After planning all needed items, they found that the quantity of rice needed for the party is 40.97 kg which is near 50 kg. Then they decided to buy 50 kg of rice instead of 49.97 kg.
  (a) Why do you think Primary 6 learners prefer to buy 50 kg instead of 49.97 kg?
  (b) Do you think both quantities: 50 kg and 49.97 kg are easily memorized?
  (c) How is rounding off useful in daily life?
  1.1 Reading and Writing Numbers Beyond 1,000,000 in words
  Activity
  1. A chart shows 999,999.
  (a) If you add 1 to 999,999, what will be your next number?
  (b) How many digits will the next number have?
  (c) Draw an abacus with 7 spikes. Starting from ones (right), fill in the
  number you get in (a) above to the last place value on the left.
  (d) What is the place value of 1 in the number you get in (a) above?
  (e) What name would you give to the number you get in (a) above?
  2. Now match the number figures to their corresponding number words.
  2,999,999 Fourteen million, one hundred forty
     thousand, two hundred nineteen.
  1,259,000    Two million, nine hundred ninety-nine
  thousand nine hundred ninety-nine.
  14,140,219    One million, two hundred fiftynine
  thousand.
  (a) Explain the steps you took when matching the numbers.
  (b) Use the same steps in (a) above to write the population of Rwanda 12,279,742 in words.
  (c) Why is it necessary to know how to write in number words?
  Example 1
  A country has a population of 5,600,002. Write the population in words.
  Solution
  Group the population in digits of threes. (5),(600),(002)
  Draw a place value table and fill in the digits.
  Write in words the values in the three-digit groups below.
  
  The population is five million, six hundred thousand two.
  Example 2
  A water tank holds 82,999,555 litres of water. Write the litres in words.
  Solution
  Eighty-two million nine hundred ninety-nine thousand, five hundred fifty-five litres.
  Study tip
  When you add 1 to 999,999, you get 1,000,000. To write 1,000,000 in words:
  First group the given digits in threes starting from the right to the left side of the whole number.
  Each group of three digits is known as a period.
  Draw a place value table and then write each digit under its place value.
  Write the value of each period in words. Then write the names of the periods to separate one period from another.
  Use a comma to separate millions from thousands and thousands from units.
  The periods that are millions, thousands and units are written without ‘s’. So, not in plural form.
  Application 1.1
  
  1. Write the following whole numbers in words:
  (a) 12,456,678 (b) 9,700,956 (c) 59,648,200 (d) 721,569,216
  2. Kalisa bought a cow at 456,700 Frw. Write the amount
  in words and explain your working out.
  3. Abeli collected 5,417,257 litres of milk from his
  farm in five months. Write the number of litres he
  collected in words.
  4. Agatha deposited 4,565,090 Frw in the bank. Write the amount she
  deposited in words.
  5. A non-government organisation spent 12,468,250 Frw on educating
  young people about the dangers of alcohol. Write the amount of money
  they spent in words.
  6. On the inauguration ceremony of a district chairperson, 4,412,567 people
  were invited. Write the number of people invited in words.
  7. A district planted 9,998,888 saplings. Write the number in words.
  1.2 Reading and Writing Numbers Beyond 1,000,000 in Figures
    Activity
  1. A chart shows 999,999.
  (a) If you add 1 to 999,999, what will be your next number?
  (b) How many digits will the next number have?
  (c) Draw an abacus with 7 spikes. Starting from ones (right), fill in the
  number you get in (a) above to the last place value on the left.
  (d) What is the place value of 1 in the number you get in (a) above?
  (e) What name would you give to the number you get in (a) above?
  2. Now match the number figures to their corresponding number words.
  2,999,999 Fourteen million, one hundred forty
  thousand, two hundred nineteen.
  1,259,000 Two million, nine hundred ninety-nine
  thousand nine hundred ninety-nine.
  14,140,219 One million, two hundred fiftynine
  thousand.
  Activity
  Match the following cards accordingly:
  
  Four million, five hundred sixty-five    1,900,999
  thousand, two hundred seventy.
  One million, nine hundred thousand,    1,300,860
  nine hundred ninety-nine.
  One million, three hundred    4,565,270
  thousand, eight hundred sixty.
  1. Explain the steps you took when matching the cards.
  2. Why is it necessary to know how to write numbers in figures?
  
  Study tip
  When writing whole number in figures:
  First, group the number words in threes starting from the right of the whole numbers.
  Write the digits of each period in numbers.
  Read and write each period separately.
  Use a comma to separate millions from thousands and units.
  Add the values to get the right figure.
  Application 1.2
  Write the following numbers in figures:
  1. Fifteen million, three hundred fifty-six thousand, four hundred thirteen.
  2. Eighty-three million, sixty- six thousand, two hundred thirty.
  3. Eight hundred million, eighteen thousand, seven hundred seventeen.
  4. A school collected four hundred fifty-six million, five hundred forty-five thousand, two hundred Rwandan francs as school fees from its learners. Write the amount in figures.
  5. Five hundred twelve million, five hundred forty-nine thousand Rwanda francs was spent by a school in a year. Write the amount in figures.
  6. Nineteen million Rwanda francs was spent by a company to print books. Write the amount in figures.
    1.3 Finding Place Value and Values of Numbers up to 7 Digits
  Activity
  1. Look at the cards below and form a number.
  
  2. Draw a place value table in your exercise book.
  3. Fill in the digits of the number 7,830,591 in the correct place values in
  the place value table.
  4. Write down the place value of each digit from the table.
  5. Use the place value table to write 7,830,591 in words.
  6. In which areas of your life can you apply place values?
  Example 1
  
  What is the place value of each digit in 8,356,421?
  Solution
  Method 1
  Count the digits in the number. There are 7 digits, now draw a place value table as shown below.
  
  The place value of 8 is millions.
  The place value of 3 is hundred thousands.
  The place value of 5 is ten thousands.
  The place value of 6 is thousands.
  The place value of 4 is hundreds.
  The place value of 2 is tens.
  The place value of 1 is ones
  
  Study tip
  The place values of numbers are the positions of Ones, Tens, Hundreds,Thousands, Ten thousands, Hundred thousands, Millions, Ten millions and Hundred millions.
  To find the place values of digits, draw the place value table and place the digits in it from the right.
  To find the value of digits, find the product of the digits and their place values.
  Application 1.3
  
  1. Write the place value of each digit in the following:
  (a) 2,312,983 (b) 7,676,405 (c) 10,101,899 (d) 853,925,732
  2. Write the place value of the underlined digits in the following:
  (a) 3,459,874 (b) 356,295,712 (c) 7,879,631 (d) 54,382, 345
  3. Write the value of each digit in the following:
  (a) 24,567,400 (b) 894,500,678 (c) 208,567,120 (d) 120,394,456
  4. Write the value of the underlined digits in the following:
  (a) 34,475,576 (b) 687,034,239 (c) 126,687,893 (d) 653,592,721
  5. Describe how you can identify the value of 6 in 4,567,890
  1.4 Comparing Numbers Using <, > or =
  (a) Which one is bigger than the other?
  (b) Explain your answer.
  (c) Complete the sentence using either <, > or =. That is A ...... B.
  2. Count the total number of learners in your class.
  (a) Use <, > or = to compare the number of boys to girls in your class.
  (b) How can you apply comparison in your daily life situations?
  Example 1
  Compare 6,312,542 and 6,312,452 using <, > or =.
  Solution
  Draw a place value table and fill in the numbers.
  
  Compare the digits in each place value from left to right.
  6 = 6, 3 = 3, 1 = 1, 2 = 2, 5 > 4 in the hundreds of units place value.
  Therefore, 6,312,542 > 6,312,452
  Example 2
  Compare 42,635,989 and 42,543,129 using <, > or =.
  Solution
  Draw a place value table and fill in the numbers.
  Compare the digits in each place value from left to right.
  4 = 4, 2 = 2, 6 > 5 in the hundred thousands place value.
  Therefore, 42,635,989 > 42,543,129.
    Example 3
  Imanirere sold clothes worth 2,560,320 Frw in 2015. She sold clothes worth 4,576,670 Frw in 2016. Compare the sales over the two years using <, > or=.
  Solution
  Draw a place value table and fill in the numbers.
  Compare the digits in each place value from left to right.
  2 < 4 in the millions place value.
  Therefore 2,560,320 < 4,576,670
  Study tip
  The symbol > means greater than, < means less than and = means equal to.
  Compare numbers using place values. Check the biggest number against the smallest number.
  You can compare numbers by counting the number of digits of the whole numbers given.
- For numbers with the same number of digits, compare the digits with the same place value from the left. The number with the bigger digit is greater than the other. The number with the smaller digit is lesser than the othe
- For numbers with a different number of digits, the one with the highest number of digits is greater that the one with the lowest number of digits.

Key unit competence: To be able to multiply and divide integers.

Introduction
In Mathematics, operations on integers are performed in a range of situations.
Suppose you are moving and counting from point A to point B with strides
forward for a distance of 20 strides.
(a) What happens if you reach the point B and go back jumping 2 strides four
times without changing the direction you are facing? Is the movement
positive or negative?
(b) How many jumps are needed to cover the distance from point B to A?
Give the mathematical operation used to get the answer.
(c) Consider the starting point A as zero. What will happen if you continue
jumping and pass the starting point A four strides?

2.1 Multiplying Integers Using a Number Line

Activity

Draw a number line on the ground.
Jump from 0 to 4, then from 4 to 8 and from 8 to 12.
How many jumps have you made?
Record your findings and discuss.
What do you notice?

Study tip

When multiplying integers using a number line, move intervals which are

equal to the multiplied number, times the number of the multiplier.

Multiplication is like repeated addition.
To multiply a positive integer by a positive even or odd number, move to the positive side.

To multiply a negative integer by a positive odd or even number, move to the negative side.

To multiply a negative integer by a positive odd or even number, move to the negative side.

Application 2.1

1. Use a number line to multiply the following:
(a) (+8) × (+5) = (b) (+11) × (-2) = (c) (+6) × (4) =
(d) (+3) × (9) = (e) (+12) × (+3) = (f) (+5) × (4) =
2. Kahesi makes a stride of 3 gaps on a number line to its right. Find the total
integers he makes in 6 strides.
3. Gashumba made 9 jumps. Each is of 2 gaps on a number line to its left.
Write an expression that describes the statement. What is the answer?

2.2 Multiplying Integers without Using a Number Line

Activity

Pick number cards that complete the following statements:
(a) 3 x +6 = (b) 4 x 9 = (c) +8 x +6 = (d) +2 x 6 =
36 12 48 18
Explain the steps taken to arrive at the answer.
Make a presentation to the class.

Example

Multiply the following without using a number line:
(a) +7 × +9 = (b) −3 × +12 = (c) +5 × −8 = (d) −7 × −8 =
Solution
(+ × + = +) (− × + = −) (+ × − = −) (− × − = +)
(a) +7 × +9 = +63 (b) −3 × +12 = −36 (c) +5 × −8 = −40 (d) −7 × −8 = +56

Study tip

Multiplying integers with the same sign gives a positive. That is, positive ×

positive gives a positive, and negative × negative gives a positive.Multiplying integers with different signs gives a negative. That is, positive × negative gives a negative and negative × positive gives a negative.

Application 2.2

Work out the following:
(a) +4 × +4 = (b) −5 × −8 = (c) +10 × −7 = (d) −9 × +6 =
(e) 12 × −8 = (f) −4(+3) = (g) −11 × −9 = +7 × +6 =
(i) −24 × −25 = (j) 120 x −5 = (k) −125 × 12 = (l) −123 × −101 =

2.3 Dividing Integers Using a Number Line

Activity

Draw a number line on the ground.
Jump from +28 to +21, then from +21 to +14, then from +14 to +7 and finally from +7 to 0.
How many jumps have you made?
Record your findings and discuss.
What do you notice?

Study tip

Division is like repeated subtraction.
Start from zero and make intervals equivalent to the divisor in length until you reach the dividend.
The number of intervals becomes the answer.
If the dividend is in the positive direction (quotient), the integer is positive.
If the dividend is in the negative direction, the integer is negative.

Application 2.3

Use number lines to divide the following:

(a) +20 ÷ +5 = (b) −16 ÷ +2 = (c) +18 ÷ −3 = (d) −28 ÷ −4 =
(e) +12 ÷ −3 = (f) +27 ÷ +3 = (g) −9 ÷ −3 = −24 ÷ +8 =

2.4 Dividing Integers without Using a Number Line

Activity

Pick number cards that complete the following statements:
(a) (+18) ÷ (−3) = (b) (+77) ÷ (+7) =
(c) (−50) ÷ (10) = (d) (+12) ÷ (−3) =
−5 −4 −6 +11
Explain the steps taken to get the answer.
Make a presentation to the class.

Example

Divide the following without using a number line.

Divide the following without using a number line.
(a) +12 ÷ +3 = (b) −21 ÷ +7 = (c) −48 ÷ − 8 =
Solution
(+ ÷ + = +) (− ÷ + = −) (− ÷ − = +)
(a) +12 ÷ +3 = +4 (b) −21 ÷ +7 = −3 (c) −48 ÷ − 8 = +6

Study tip

A positive integer divided by a positive integer the answer is a positive integer.
A positive integer divided by a negative integer the answer is a negative integer.
A negative integer divided by a negative integer the answer is a positive integer.

Application 2.4

Divide the following without using a number line.
(a) (+39) ÷ (-13) = (b) (-21) ÷ (-7) = (c) (-50) ÷ (+10) =
(d) (+44) ÷ (+11) = (e) (-18) ÷ (-6) = (f) +24 ÷ +8 =
(g) −32 ÷ −4 = −96 ÷ −8 = (i) −125 ÷ −25 =
(j) +625 ÷ +125 = (k) −3,500 ÷ +700 = (l) +299,160 ÷ +12465 =

2.5 Solving Problems Involving Multiplication and Division of Integers

Activity

Think of a number.
Divide it by +4.
Multiply the answer by -2.
If the final answer is 6 what is the number?
Defend your answer.

Example 1

Work out the following:

Work out the following:
Temperatures of an area increases by +4⁰C per hour during the day. What is
the total increase in 6 hours?
Solution
1 hr = +4⁰C
6 hrs = +4⁰C × +6
= +24⁰C

EXAMPLE2

Study tip

Integers may be applied in real life situation, such as loss and profit, temperature rise and fall, ascending and descending altitude.
You can use multiplication to check your division.

Application 2.5

1. In a village, there is a famous trader called Mugisha. He bought a 100 kg sack
of rice at 230,000 Frw. How much did he pay for each kilogramme of rice?
2. Annet, Juma and Shasa shared 18 mangoes amongst themselves.
How many mangoes did each get?
3. A school spent 23,500 Frw on 47 calculators. What was the cost of
each calculator?
4. Joshua paid 15,000 Frw for milk from Bizimana’s farm during the month
of April. How much money did he pay daily?
5. Temperature in Summer increased by +3⁰C per hour. What was the
increase in 9 hours?
6. Each receipt book contains 60 debtors. Each debtor owes 24,000 Frw
to the company. Calculate the total debt.
7. The temperature change climbing a mountain was 3⁰C every after 1
decametre. A climber covered 5 decametre. What was the temperature
when he stopped?
8. Arsenal lost 3 points in each of the last 5 games played. What was the
loss altogether?

End of unit 2 assessment

1. Solve without using a number line.
(a) (+24) ÷ (+8) = (b) (-10) × (+3) = (c) (-66 ) ÷ (-11) =
(d) (-12) × (-12) = (e) (+3) × (-9) = (f) (-20) ÷ (+4)
2. Work out the following using a number line.
(a) 3 × −4 = (b) +24 ÷ −3 = (c) +30 ÷ +6 =
(d) +12 × −4 = (e) +9 × +6 = (f) +36 ÷ −4 =
3. Ngabire bought 4 kg of sugar each costing 1,200 FRW. How much
did she pay?
4. Mukahirwa owed 4 friends of hers 12,000 FRW each. She earned 40,000
FRW and paid it off to them equally. How much is she still owed by each one?
5. Share 4,800 books among 8 learners.
6. The sum of two integers is +20. Dividing the largest integer by the smallest
integer gives a quotient of -3. What are the two integers?
7. A debt of 1,600 FRW was cleared by 4 people. How much did each clear?
8. Uwera took a loan of 6,000 FRW. She paid it in 4 equal installments. How
much was each installment?
9. A debt of 1,200 FRW was cleared equally by 6 people. How much did
each person clear?
10. Write the division statement shown on the number line.

Solve without using a number line.

CLICK HERE FOR A VIDEO LESSON ON: Multiplying integers without using a number line URL

Multiplying integers without using a number line URL

Key unit competence: To be able to use powers and indices, and apply the Lowest Common Multiple (LCM) and the Greatest Common Factors (GCF) when solving problems.

Introduction

To write big numbers in figures, is sometimes challenging for some people. For example, the distance from the earth to the sun is of many millions of kilometers 149,600,000 km. To facilitate people in writing such kind of numbers, Mathematicians found a way of writing them in short form using powers or indices.

The distance from the earth to the sun can be easily written in short form as follows: 149.6 x 10⁵
(a) Have you ever come across numbers written in this form?
(b) How is writing big numbers in short form helpful? Explain.

3.1 Indices

Activity

Write 64 on a sheet of paper and prime factorise it.

(a) What common prime factor did you use?

(b) How many times did you multiply it?

Write the common prime factor once and write the number of times to its right, above it. Explain your working.

Study tip

When writing in indices, write the number that has been multiplied repeatedly once.
Count and write the number of times the number has been multiplied to the right, above that number.

Application 3.1

1. Write the following numbers using indices:

(a) 7 × 7 × 7 × 7 × 7 = (b) 6 × 6 × 6 × 6 × 6× 6 × 6 =
(c) 10 × 10 × 10 × 10 × 10 = (d) 8 × 8 × 8 =
(e) 9 × 9 × 9 × 9 × 9 × 9 × 9 = (f) 12 × 12× 12 × 12 × 12 × 12 =

2. Complete the following statements:

(a) 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4 = 4^? (b) 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^?
(c) 11 × 11 × 11 × 11 × 11 × 11 × 11 = ?⁹ (d) 20 × 20 × 20 × 20 = ?⁴

3. Write the following powers in expanded form:

(a) 3⁷ = (b) 5⁹ = (c) 2¹² = (d) 100⁶ =

3.2 Defining Base and Exponent

Activity

Study the number 7⁴ and answer the questions that follow.
(a) Which number has been multiplied repeatedly? What is its name?
(b) Which number shows the number of times the number has been multiplied repeatedly? What is its name?
Explain to the class.

Study tip

The number that is multiplied repeatedly is the base.
The number that shows the number of times a number has been multiplied repeatedly is the exponent.

Application 3.2

1. State the number that represents the base in the following:
(a) 5² (b) 6⁴ (c) 10⁷ (d) 8⁵ (e) 2⁹

2. State the exponent in the following:
(a) 7⁵ (b) 15⁹ (c) 11⁴ (d) 4⁷ (e) 18⁶

3. Given that 9 is the base and 6 is the exponent, write the number in power notation.

4. Expand 11⁷.

5. 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6. Write the number in power
State the exponent and the base.

3.3 Multiplying and the Law of Multiplication of Indices

Activity

Given 6² × 6⁵, expand and simplify.
How many times has 6 been multiplied?
Now add the indices in the first expression.
Leave your answer in power notation.
Compare your results. What do you notice?

Study tip

The law of multiplying indices: When multiplying numbers with the same base, maintain the base and add the exponents or indices.

Application 3.3

1. Simplify the following: Leave your answers in power notation.

(a) 6² × 6³ = (b) 9² × 9² = (c) 10⁵ × 10⁶ =
(d) 12⁴ × 12⁵ = (e) 4³ × 4³ × 4² = (f) 5³ × 5 =
(g) 7⁶ × 7⁴ = 20 × 20 × 20⁵ = (i) 13 ×13 =

2. Evaluate the following:

(a) 2³ × 2⁴ = (b) 10² × 10² ×10¹ = (c) 4²× 4¹ × 4³ =
(d) 5¹ × 5³×5¹ = (e) 11² × 11¹ × 11² = (f) 3 × 3³ =
(g) 2 × 2² × 2³ = 12 × 12 × 12 = (i) 1 × 1¹⁰ × 1⁴ =

3.4 Dividing and the Law of Division of Indices

Activity

Write 64 on a sheet of paper and prime factorise it.
(a) What common prime factor did you use?
(b) How many times did you multiply it?
Write the common prime factor once and write the number of times to its right,
above it. Explain your working.

Study tip

The law of dividing indices states that, when dividing numbers with the same base, you should maintain the base and subtract the exponents or indices.
Simplify means the answer is given in power notation.
Work out or evaluate means getting the value out of the expression.

Application 3.4

3.5 Multiplying and Dividing Indices

Activity

On a slip of paper, write the numbers 32, 8 and 16 in power form.
Multiply the powers of 32 by the powers of 8, then divide by the powers of 16.
What answer do you get in power form?

Study tip

When multiplying and dividing indices, use both the law of multiplying and that of dividing.

Application 3.5

3.6 Finding Unknown and the Law of Multiplying Indices

Activity

Write two multiples of 5.
Express them in power form.
Use the law of multiplying indices to find the answer in power notation.
If the index of one of the numbers is not given, and the second is given, use the answer to find the unknown index.

Present your answer to the class.

Study tip

To find the unknown index, form an equation using the law of multiplying indices. Then solve it.

Application 3.6

3.7 Finding the Unknown and the Law of Dividing Indices

Activity

Write two numbers in power form.
Form a statement of dividing and the law of dividing indices for the above numbers.
Explain your working to the class.

Study tip

To find the unknown index in dividing and the law of dividing indices, form an equation of the indices, then solve it.

Application 3.7

3.8 Finding the Lowest Common Multiple (LCM) of Numbers

Activity

Write 8 and 12 on slips of paper.

Write down the first ten multiples of each number.
Identify their common multiples.
What is their lowest common multiple?

Study tip

To find the LCM, either list the multiples of the given numbers and choose the lowest common multiple, or prime factorize the given numbers then find the product of all the prime factors.
The Lowest Common Multiple (LCM) is also known as the Lowest Common Denominator (LCD) in case of denominators of fractions.

Application 3.8

Find the LCM of the following:

(a) 8 and 6    (b) 18 and 16    (c) 7 and 11
(d) 48 and 32    (e) 10, 12, and 15    (f) 20, 25 and 30
(g) 25, 30 and 45    45, 60, and 70

3.9 Solving Problems Involving LC

Activity

Two alarm clocks ring in intervals of 10 and 15 minutes.
Make a list of time intervals for each of the two clocks.
What is the first common time interval of the two clocks?
What does the first common time interval mean?
Present your findings to the class.

Example

Mugisha was diagnosed with malaria and the doctor prescribed the treatment as follows: 4 tablets of Coartem to be swallowed every 12 hours and 2 tablets of Cotrimazole to be swallowed every 8 hours. How long will Mugisha swallow both drugs together again?

Solution

Multiples of 12 are 12, 24, 36, 48, 60, 72, ...

Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, ...

Common multiples are 24 and 48.

LCM is 24. Therefore, Mugisha will swallow both medicines together again after 24 hours.

Study tip

To solve problems involving LCM, list the multiples of the given numbers then choose the lowest common multiple.
Read and understand the question thoroughly.

Application 3.9

Work out the following:
1. Three taxis leave the park at intervals of 15, 20 and 25 minutes. After
how long will the taxis leave the park at the same time?
2. Ganza packed books in boxes that carry 15 books each. Ines could not
carry the boxes because they were too heavy for her. She repacked the
books in boxes carrying 9 books each. How many books were there if
she did not leave any books out?
3. Two nurses administer medication to two patients in intervals of 30 and
45 minutes respectively. How long will the nurses administer medication
to the patients at the same time if they
started at the same time?
4. Three buses arrive at a bus park at
intervals of 30, 40 and 45 minutes.
How long will the buses take to arrive
at the park at the same time if their
first arrival time was the same?
5. Kamana plants maize and groundnuts
in spaces 12 cm and 15 cm apart
respectively. At what distance will both plants be planted together?

3.10 Finding the Greatest Common Factor (GCF) of numbers

Activity

List 3 numbers of your choice.
Find the factors of each number.
List the factors that are common.
What is the greatest of all the common factors?
Present your working out to the whole class.

Study tip

To find the greatest common factor, list all the factors of the given numbers. Pick out the common factors then the greatest common factor.
You can also use a factor tree to find the common factors, then work out their product.

Application 3.10

Find the greatest common factor (GCF) of the following:

1. Use listing of factors.

(a) 30 and 40 (b) 120 and 180 (c) 180 and
(d) 60, 120 and 180 (e) 120 and 60 (f) 45, 60, and 75

2. Use prime factorization.

(a) 40 and 64 (b) 64 and 128 (c) 42, 56, and
(d) 120, 220, and 360 (e) 48, 72 and 96 (f) 50, 75, and 90

3.11 Solving Problems Involving GCF

Activity

Two wires 448 cm and 616 cm in length are to be cut into pieces of the same length without any remainder.
What do we do to find the greatest possible length of the pieces?
Try to work out the problem.
Present your findings to the class.

Example

Joel fetched 60 litres of water in the morning and 72 litres in the evening. Find the capacity of the biggest container Joel used in both instances.

Solution

The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
The common factors are: 1, 2, 3, 4, 6, 12.
The greatest common factor is 12.
The biggest container Joel can use in both instances has a capacity of 12 litres.

Study tip

GCF is useful in packaging products in factories.
GCF is useful in simplifying fractions to the lowest term.
GCF is used in dividing two or more numbers without leaving a remainder.

Application 3.11

1. What is the greatest number that can divide 36 and 54 without leaving
a remainder?
2. Mutesi collects 30 litres of milk from her farm in
the morning. She collects 35 litres of milk in the
afternoon and 45 litres in the evening. What is
the capacity of the biggest container that can
be used in all instances with no milk remaining?
3. Musa collected 48 kg of okra seeds from one
garden and 84 kg from another. Find the mass
of the pack that can be used in both instances
without leaving any okra seeds in the garden.
4. The workers at a maize mill have 3 sacks of maize flour weighing 90 kg,
120 kg and 150 kg respectively. What is the mass of the biggest pack
that can be used so that no flour remains in any of the sacks?
5. What number when divided by 6, 8 and 12 gives 3 no remainder?

3.12 Finding the Unknown Number Using LCM and GCF

Activity

Write two numbers of your choice on slips of paper.
Work out their LCM.
Now find the GCF of the two numbers you chose.
Calculate the product of the two numbers, then divide by the LCM. What do you observe?
Now calculate by dividing the previous product by the GCF.
What is the relationship between the product, GCF and LCM?

Study tip

The product of two numbers is equal to the product of the GCF and LCM.
LCM is equal to the product of two numbers divided by the GCF.
GCF is equal to the product of two numbers divided by the LCM.
The unknown number is equal to the product of the LCM and GCF divided by the given number.

Application 3.12

1. The LCM of two numbers is 24. Find the second number if one of the numbers is 8.
2. The GCF of two numbers is 16. The numbers are 48 and x. Find x.
3. The two numbers are 9 and 15. What is their GCF and LCM?
4. The LCM of two numbers is 180. GCF is 25, one of the numbers is 50. What is the second number?
5. Calculate the GCF of 45 and 60.
6. The LCM of two numbers is 144. The GCF is 2. If one of the numbers 18, what is the other
7. The GCF of two numbers is 3. The LCM is 60. If one of the numbers 12, find the second
8. One of the numbers in a pair is h. The other number is 8. If the LCM 24 and the GCF is 4, what is the value of h?

End of unit 3 assessment

1. Work out the following:
(a) 8⁴ ÷ 8³ = (b) 3³ × ..... = 3⁵ (c) ..... ÷ 7² = 7³
(d) 5⁵÷.. = 5² (e) 12³ × 12-² = ? (f) 13⁴ × ... ÷ 13² = 13⁵
2. Find the LCM and GCF of the following:
(a) 60 and 120 (b) 36, 48 and 92 (c) 9, 15 and 36
3. John and Jesca were each given sugarcane of equal length. John cut his
sugarcane into equal lengths of 20 cm, while Jesca cut her sugar cane into
equal lengths of 50 cm. If they don’t have any remainder, find the shortest
possible length of sugarcane should each equally have
4. 3 bells at a factory ring at intervals of 15 minutes, 20 minutes and 30
minutes to mark shifts of different departments. If they first ring together
at 8:00 am. When do they ring together again?
5. Uwera prepared 96 litres of orange juice, 72 litres pineapple juice and 84
litres of watermelon juice she kept it small containers. What is the capacity
of the biggest container she used if no juice remained?
6. At exactly 6:00 a.m, two buses leave the bus terminal. Thereafter, every
40 and 70 minutes, a bus leaves the terminal. Find the time the buses will
leave the terminal together.

Key unit competence: To be able to apply fractions in daily life situations and solve related problems.

Introduction

In real life, people share many things. They share food, drinks, money, land and so many other things. When people are sharing, each person receives a portion of a whole and this portion is known as fraction in Mathematics.Suppose that a pineapple is shared equally among 3 people.

(a) What portion/fraction does each person get?

(b) If two of them decide to combine their portions, what mathematical operations do they carry out?

(b) Find out some examples of equal sharing involving Mathematical operations in real life, and present it using fractions.

4.1 Multiplying a Whole Number by a Fraction

Activity

Collect 9 halves of beans.
Write one part as a fraction of a whole bean.
Form a multiplication statement of the number of halves of the beans multiplied by a half of a bean.
Explain your working out to the class.

Example 1 Example 2

multiply

Example 1 Example 2

Study tip

To multiply a whole number by a fraction, multiply the whole number by the numerator, then divide the product by the denominator. Simplify where necessary.

Application 4.1

$multiplication of fractions$

4.2 Multiplying a Fraction by a Whole Number

Activity

Look at the sugarcane below:

1. How many parts does the sugarcane have?

2. Write the fraction representing one part.

3. Multiply the fraction above with the number of parts.

4. What do you get?

Example 1

$fraction$ $fraction$

Study tip

To multiply a fraction by a whole number, multiply the numerator by the whole number. Divide the product by the denominator and simplify where possible.

Application 4.2

$fractions$

4.3 Multiplying a Fraction by a Fraction

Activity

Draw a shape of a whole.

Divide it into 10 equal parts.

Shade 5 parts.

Now divide each part into halves.

Shade 1/2 of the shaded part.

Count and form a fraction of the double shaded part out of all the parts of the whole. What is your answer?

Now multiply 1/2 by 5/10. What do you get? Present your findings to the class.

Example 1 Example 2

$fraction$ $fraction$

Study tip

When multiplying a fraction by a fraction, multiply both numerators separately and then the denominators separately. Simplify where possible.

Application 4.3

Multiply the following fractions.

$multiplication of fraction$

4.4 Solving Problems Involving Multiplying Fractions

Activity

Mbabazi and Rugari stood for the post of Chairperson for their saving group.Mbabazi got 1/3

of the votes and Rugari got the rest.

If 3,600 people voted:

(a) How many people voted for Mbabazi?

(b) What fraction voted for Rugar?

Mary is a businesswoman. Her income in a year is 12,000,000 Frw. She calculated her business expenditure and found out that 1/5 of her income is spent on advertisement of her products.

She spends 3/20 of her income on paying her staff.

The goods that she bought cost 1/3

of her income. She saved the rest.

1. How much money does Mary pay for advertising?

2. How much money does Mary pay her staff?

3. What is the cost of the goods that Mary buys?

4. How much does she save?

Example 2

Uwera had land. She gave 2/5 of it to her son, Kwizera. She also gave 3/8 of the land to her daughter, Cissy.

(a) What fraction of the land did she remain with?

(b) If Uwera remained with 18 hectares, calculate the hectares she had originally.

Application 4.4

1. Jasmine had 300,000 Frw in her purse. She gave 3 /10 of it to Jane, 1/5 to Julian and saved the rest.

(a) What fraction of the money did she save?

(b) How much money did Jasmine save?

2. Three men shared 800 kg of beans. Ali got 1/4 of the beans, Moses got 3/8 and Katto got the remaining beans.

(a) Calculate the fraction Katto got.

(b) Work out the amount of beans in kilogram each got.

3. A painter painted 1/5 of the room with white colour and 4/ 10 of it with blue

colour. He then painted the remaining part with 5 litres of green colour.

(a) What part was painted green?

(b) Work out the amount of paint the painter used to paint the entire classroom.

4. Moses spent 1/4 of his money on food. He also spent 1/3 of the remaining money on transport. He was left with 12,000 Frw. How much money did he have originally?

5. Kanyange planted 1/8 of her land with sorghum, 1/4 of it with maize and 1/2 of the remaining land with beans. She planted other crops on the remaining sacres. How much land did she have in acres?

4.5 Finding Reciprocals

Activity

Write a number and multiply it by any unknown.
Equate it to one (1) to form an equation.
Calculate the value of the unknown.
What do you observe about the number you wrote at first compared to the result?
Present your findings to the class.

Study tip

The reciprocal of a given number is the other number that is multiplied by it to get a product (1).
The reciprocal of a whole number is a fraction.
The reciprocal of a fraction is a whole number.

Application 4.5

4.6 Dividing a Whole Number by a Fraction

Activity

Collect 2 sheets of paper.
Cut each in four equal parts.
What fraction of a whole sheet is each portion?
Count all the portions altogether.
Formulate a Mathematical statement for the activity.

Study tip

To divide a whole by a fraction, multiply the whole by the reciprocal of the fraction.
Simplify where necessary.

Application 4.6

4.7 Dividing a Fraction by a Whole Number

Activity

Draw an orange and divide it into two equal parts.
Divide each portion into 3 portions.
Now count the portions you have.
What is the fraction of one portion out of all the portions?
Present your findings to the class.

Study tip

To divide a fraction by a whole number, first write the fraction, then multiply the fraction by the reciprocal of the whole number.

4.8 Dividing a Fraction by a Fraction

Activity

Draw a circle and divide it into 8 equal portions.
Shade 3 portions and write down the fraction of the shaded part.
Divide the shaded portion into 2 equal parts.
Shade one portion. Write it as a fraction of the first shaded portion.
Divide the circle into more equal small portions.
How many are there?
What is the relationship between the first and final shaded parts?
What is the fraction of the final shaded portion out of all the small portions of the circle? Write it down.
Share the procedure with the whole class.

Study tip

When dividing a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction. Simplify where possible.

4.9 Solving Problems Involving Dividing Fractions

Activity

Juma collected 15 litres of water which is three quarters of a jerrycan. What is the capacity of the container Juma used? Present your working to the class.

Example

Study tip

In order to work out problems involving dividing fractions, read and interpret in order to understand.
Identify the operations to be used. The given portion is divided by the given amount to find the total quantity.

4.10 Multiplying and Dividing Fractions

Activity

Write three fractions of your choice.
Multiply the first two.
Divide the product of the first two by the third fraction.
Which steps did you carry out to get the answer?
Present your working to the class.

Example 1

Example 2

Study tip

When calculating combined multiplying and dividing fractions, work out division first, then multiply. Use BODMAS.
Multiplication and division of fractions can be worked by changing the divided fraction to its reciprocal. Then the fractions are multiplied together.

Key unit competence: To be able to round off decimals, convert fractions to decimals and vice versa.

Introduction

In counting and mathematical operations, some numbers may be found difficult to be memorized or used. These can be simplified to their nearest numbers which can be easily memorized or used. For example money used by different schools per year can be rounded to simple figures to summarize.

Let’s consider an example of school party where Primary 6 learners contributed to buy all items needed. After planning all needed items, they found that the quantity of rice needed for the party is 40.97 kg which is near 50 kg. Then they decided to buy 50 kg of rice instead of 49.97 kg.

(a) Why do you think Primary 6 learners prefer to buy 50 kg instead of 49.97 kg?

(b) Do you think both quantities: 50 kg and 49.97 kg are easily memorized?

5.1 Rounding off Decimal Numbers to the Nearest Tenth

Activity

Pick a flash card from A and match it with its corresponding card in B
containing a value nearest to it.

What have you noticed?
Present your findings to the class.

Study tip

To round off to the nearest tenth, first identify the place value of tenths.Then find the digit to its right..
If it is 0, 1, 2, 3 or 4, add 0 to the digit in the tenths place value. This is because the digits are nearest to 0.
If it is 5, 6, 7, 8 or 9, add 1 to the digit in the tenths place value. This is because the digits are nearest to 10.
The rounded off answer must be to one decimal place.

Application 5.1

5.2 Rounding off Decimal Numbers to the Nearest Hundredths

Activity

Read, think and respond quickly, then explain.
1. 123.1234 rounded to the nearest hundredth is equal to….
2. 239.5589 rounded to the nearest hundredth is equal to….
3. Explain the difference between those two numbers.
4. Present your findings to the class.

To round off to the nearest hundredth, first identify the place value of hundredths and find the digit to its right.

If it is 0, 1, 2, 3 or 4, add 0 to the digit in the hundredths place value. If it is 5, 6, 7, 8 or 9, add 1 to the digit in the hundredths place value.
The rounded off answer must be to two decimal places.

Application 5.2

Round off the following decimals to the nearest hundredth:
(a) 16.597    (b) 4.221 (c) 0.571    (d) 8.009
(e) 367.812    (f) 6.0027    (g) 5.2743 0.6451
(i) 7.0095    (j) 4.3214    (k) 6.7890    (l) 9.248

5.3 Rounding off Decimal Numbers to the Nearest Thousandth

Activity

Pick a flash card from A and match it with its corresponding card in B
containing a value nearest to it.

What have you noticed?
Present your findings to the class.

Study tip

To round off to the nearest thousandth, first identify the thousandths place value.
If the digit to the right of the thousandths place value is 0, 1, 2, 3, or 4, add 0 to the digit in the digit in the thousandths place value.
If the digit to the right of the thousandths place value is 5, 6, 7, 8, or 9, add 1 to the thousandths place value.
The rounded off answer must possess three decimal places.

Application 5.3

Round off the following to the nearest thousandth:
(a) 2.7985    (b) 12. 3421    (c) 5.6874 (d) 125.8213
(e) 1.4685    (f) 25.0098    (g) 7.23567    34.69745
(i) 0.69736    (j) 67.7983    (k) 89.8357 (l) 295.2110

5.4 Rounding off Decimal Numbers to the Nearest Ten Thousandth

Activity

Study the number cards below:

7.04683 0.62437 1,625.00016 0.99685

Write the approximate values of the numbers estimated to ten thousandth.
Explain the procedure to the class.

Study tip

To round off to the nearest ten thousandth, first identify the ten thousandths place value.

If the digit to the right of the ten thousandths place value is 0, 1, 2, 3, or 4, add zero (0) to the digit in the ten thousandths place value.

If the digit to the right of the ten thousandths place value is 5, 6, 7, 8 or 9, add 1 to the digit in the ten thousandths place value.

The rounded off answer must be to four decimal places.

Application 5.4

Round off the following to the nearest ten thousandth:
(a) 0.06951    (b) 12.106798    (c) 482.00311    (d) 0.002456
(e) 4.99999 (f) 19.00283    (g) 0.97568 73.01010
(i) 1,206.07989    (j) 63,006.7099 (k) 723.909    (l) 0.917801

5.5 Rounding off Decimal Numbers to the Nearest Hundred Thousandth

Activity

Study the decimal numbers in A and B.

Match the corresponding decimal numbers in A to those in B.
Defend your answers.

Study tip

To round off to the nearest hundred thousandth, first identify the place value of hundred thousandth, then find the digit to its right.

If it is 0, 1, 2, 3, or 4, add 0 to the digit in the hundred thousandths place value.
If it is 5, 6, 7, 8 or 9, add 1 to the digit in the hundred thousandths place value.
The rounded off answer must be to five decimal places.

Application 5.5

Round off the following to the nearest hundredth thousandth:
(a) 1.1010110    (b) 0.063285    (c) 0.006192
(d) 15.73912    (e) 92.093018    (f) 0.185306
(g) 100.683048    9.999999    (i) 45.789093

5.6 Rounding off Decimal Numbers to the Nearest Millionth

Activity

Pick five cards with 7 and 8 decimal places from the pack.
Study each card critically.
On slips of paper, write the approximate of each number to six decimal places.
What did you consider when approximating?

Study tip

To round off to the nearest millionth, first identify the place value of millionth and find the digit to its right.

If the digit is 0, 1, 2, 3 or 4, add 0 to the digit in the millionth place value.
Add 1 to the digit in the millionth place value if the digit is 5, 6, 7, 8 or 9
The rounded off answer must be with six decimal places.

Application 5.6

Round off the following decimals to the nearest millionth.
(a) 0.1251076 (b) 4.1407324 (c) 0.0506985
(d) 13.1303643 (e) 1.1109427 (f) 46.9328342
(g) 0.8342071 0.0321969 (i) 85.0732804

5.7 Solving Problems Involving Rounding off Decimal Numbers

Activity

Measure the length, then the width, of the football pitch.
Note the measurements on slips of paper.
(a) What is the area to the nearest whole? (ones)
(b) Round off the width to the nearest tenth.
Discuss how rounding off can be applied in daily life.

Study tip

Read and understand the statements.
Consider rounding off when estimating quantities.
If the digit in the place value to the right of the required place value is 0, 1,2, 3, or 4, round down.
If the digit in the place value to the right is 5, 6, 7, 8, or 9, round up.

Application 5.7

1. A trader bought beans weighing 1,672.3 kg. Approximately how many
kilograms were there when rounded to the nearest one?
2. Richard’s height is 177.3 cm. Write this height to the nearest whole number.
3. In a school store, there are 362.51 kg of beans and 506.78 kg of maize
flour. Round off the total kilograms to the nearest tenth.
4. Michael Jordan ran 100 m in 9.83 seconds. Round off the time to the
nearest second.
5. A forest has an area of 6.234 hectares. Round off this area to th e
nearest hundredth.
6. At birth, Winfred weighed 3.27 kg. Round off the weight to the nearest kilogram.
7. The area of a district is 1,467.49 hectares. Round off the area to the
nearest tenth.
8. The distance between two towns is 236.17 km. Round off the distance
to the nearest kilometer.

5.8 Converting Fractions into Decimals

Activity

Look at the fractions in part A and decimals in part B. Match them accordingly.
Explain your matching process to the whole class.

Study tip

When converting fractions into decimals, we divide the numerator by the denominator.
To change a fraction to a decimal, take a number which you can multiply by the denominator to get multiples of 10, 100, 1000, etc.
Multiply the number by both the numerator and the denominator. Then convert to decimals considering the place value of the denominator.
For mixed fractions like in example 3, first change the mixed fractions to an improper fraction, then divide the numerator by the denominator.
When a fraction gives a decimal number with repeating digits, the decimal number is called a recurring decimal. It is written with an ellipsis to the right.

Application 5.8

5.9 Converting Decimals into Fractions

Activity

Write 3 decimal numbers:
~ The first with 1 decimal place.
~ The second with 2 decimal places.
~ The third with 3 decimal places.
Ignore the decimal point and write the denominator under each decimal, the
zeros matching the number of decimal places.
What is your observation?

Study tip

When converting decimals into fractions, simply ignore the decimal point and write

all digits of decimal number as the numerator, then the denominator depends on

the number of decimal place. To know the denominator, you count the number of digits to your right after the decimal places.

If the decimal place is 1, the denominator is 10.
If the decimal places are 2, the denominator is 100.
If the decimal places are 3, the denominator is 1,000.
If the decimal places are 4, the denominator is 10,000.
If the decimal places are 5, the denominator is 100,000.
If the decimal places are 6, the denominator is 1,000,000.

If a decimal fraction recurs in the tenth place value, equate it to unknown to form an algebraic expression. Multiply both sides by 10 to form a second expression,subtract the first expression from the second expression. If a decimal recurs in the hundredth, thousands multiply by 100 and 1000 respectively then simplify.

Application 5.9

Convert the following decimals into fractions and simplify completely.

(a) 0.1234 (b) 19.67 (c) 195.008 (d) 10.08

(e) 17.3 (f) 13.5 (g) 54.029 0.375

(i) 0.256 (j) 56.35 (k) 6.18 (l) 25.25

(m) 25.3 18.75 (o) 0.875 (p) 0.3...

(q) 0.5... (r) 0.45... (s) 0.125... (t) 0.09...

5.10 Solving Problems Involving Converting Decimals into Fractions and Fractions into Decimals

Activity

~ Think of a decimal number with three decimal places.
~ Write it on a piece of paper.
~ Give it to your neighbour.
~ Ask him/her to round it to the tenths place value and convert it into a fraction.
~ Discuss the whole process.
~ Is the final result the same as the first? Explain.

Study tip

When solving problems, read and try to understand the problem then choose

how to solve it.

Application 5.10

Work out the following:
1. Bwiza scored the following marks in different Mathematics tests: 89.5%, 73.25% and 69.5%. How many marks did she score altogether? What is her average mark as a fraction?

2. If 58 out of 100 text books in the school are mathematics, write a decimal for the portion of the school text books which are Mathematics.

3. A tailor wants to cut 24 small pieces of 0.4 m from a roll of cloth. What is the length of the roll of cloth as a fraction?

4. Kanyana cooked 0.25 kg of rice for lunch. What is the amount of the rice she cooked as a common fraction?

5. A taxi covered a distance of 72.03 km. What is the

distance covered by the taxi as a common fraction?

6. The distance from Kalisa’s farm to his home is 12 km 200 m. Express the distance as a decimal.

7. Musa, Sarah and Akida weigh 36.7 kg, 41.3 kg and 57.6 respectively. Express their average weight
as a fraction.

8. When the circumference of a circle is divided by its diameter, the result is 3.14. Express this as a fraction.

Introduction

In daily life people compare quantities, share proportionally and mix things or objects for different reasons. Do you ever wonder why knowing the following mathematical concepts (ratio, percentage, proportion, mixture) is necessary?

For example look at your classmates:

(a) Find out the number of boys, then the number of girls. Can you express the number of boys or girls in terms percentage?
(b) Can you equally share a certain number of Mathematics textbooks to different groups in your classroom and then figure out the ratio of Mathematics textbooks per learner?
(c) Give examples where the concepts of ratios, percentages and mixtures are used in real life.

6.1 Converting Percentages into Decimals

Activity

You have been scoring marks in test and examinations.
Write the percentages on slips of paper.
Express the % mark as a denominator of 100.
Write each percentage value out of 100.
Considering the number of zeros in 100, write decimals with places of the same number.
What are your answers?
Make a class presentation.

Study tip

Application 6.1

6.2 Converting Decimals into Percentages

Activity

Formulate five decimal numbers of 1, 2, and 3 decimal places.
Express them as fractions, but do not simplify.
Multiply each fraction by 100.
What are your answers?

Study tip

To convert decimals into percentages, change the decimal into a fraction, then multiply the fraction by 100%.
Remember to put the percentage (%) mark to the right side.
To covert decimals into percentages, you can also move the decimal point two places to the right. This is because 100 has 2 zeros, hence 2 decimal places.
Then add a % symbol.

Application 6.2

6.3 Converting Percentages into Fractions

Activity

Write four percentages on a slip of paper.
Express them as fractions of the denominator 100.
Simplify each fraction.
What do you get?

Study tip

To change percentages into fractions, make the given figure the numerator and 100 the denominator, then reduce where possible.
To change fractional percentages into common fractions, divide the fractional part by 100. Then multiply by the reciprocal of 100.

Application 6.3

6.4 Converting Fractions into Percentages

Activity

Study tip

When converting a fraction into a percentage, multiply the fraction by 100%.
Then simplify to the simplest terms.
The percentage obtained is equivalent to the fraction.
Percentages are fractions with the denominator 100.

Application 6.4

6.5 Comparing Quantities as Percentages

Activity

Convert the following into percentages:

(a) 6 pencils out of 1 dozen pencils.

(b) 30 minutes out of 1 hour.

(d) 250 metres out of 2 kilometres.

Present your working to the class.

Study tip

When comparing quantities as percentages, first express the quantity as a fraction then multiply the fraction formed by 100%.
The total is always 100%. To find the original number, multiply the value of the unit percentage by 100%.
When comparing quantities as percentages, first convert the units to be the same. Express as a fraction, then multiply the fraction by 100%.

Application 6.5

6.6 Comparing Percentages as Quantities

Activity

Write 10% on slips of paper.
Let 10% represent 20 books.
Form groups of 10% equating them to groups of 20 books.

10% is equivalent to 20 books.
20% is equivalent to 40 books.
30% is equivalent to 60 books.
40% is equivalent to 80 books and so on up to 100%
.........................................................................................
.........................................................................................
100% is equivalent to ? books.
100% is taken to be equivalent to total books.

How many books are there?
Present your working to the class.

Study tip

Divide the given quantity by the given percentage to get the equivalent of 1%.
Multiply that fraction by 100% to get the total quantity/amount.

Application 6.6

6.7 Increasing a Number by a Percentage

Activity

Dusabe had 1,000 Frw.
Her father increased it by 20%. How much does she have now?
20% increase means:

Each 100 Frw, the increase is 20 Frw.
   200 Frw, the increase is 40 Frw.
   300 Frw, the increase is 60 Frw.
   ............., the increase is ? Frw.
   .............................................…
   1,000 Frw, the increase is ? Frw.

Add the increase to 1,000 Frw.
What is your answer?

Study tip

To increase a number by a given percentage, add the given percentage to 100%, then multiply by the given number.

To get the increased new amount, work out the increased amount, then add it to the old amount.

Increasing a given number by a given percentage means, adding the quantity on each 100 of the original quantity.

Application 6.7

6.8 Decreasing a Number by a Percentage

Activity

- If an item costs 500Frw.
- Its price was decreased by 5%. What is new price now?
Each 100Frw, the decrease is 5Frw.

200Frw the decrease is 10Frw.

..... Frw the decrease is .... Frw.

.....Frw the decrease is ....Frw.

-Subtract the total decrease from the original price.

- What is the new price?

Study tip

-To decrease a number by a given percentage, subtract the given percentage from 100%, then multiply by the given number.

-To get the new amount after decreasing, work out the decrease first, then subtract it from the given amount.

Application 6.8

1. Decrease 800 by 12%.
2. Decrease 960 by 30%.
3. Decrease 1,500 by 24%.
4. 400 kg is decreased by 16%, what is the new amount?
5. The marked price of a new car was 4,000,000 Frw. It was reduced by 6%. At what price was it to be sold?
6. A trader bought 650 kg of beans at 600 Frw per kg. The price decreased by 8% after the harvest season. How much did she get after the sale?
7. Decrease 101,000 kg by 30% then by 10%.
8. The marked price of a motorcycle was 1,000,000 Frw. It was decreased by 12% then by 9%. What is the new price?

6.9 More about Increasing and Decreasing Quantities by Percentage

Activity

- Think of an amount of money. Write it on slips of paper.

- If the amount is increased by 40%, it becomes 28,000 Frw.

- What was the old amount?

- Explain your working to the class.

Study tip

To find the unknown number, find 1% of the increased amount then multiply by 100.
To find the unknown number, find 1% of the decreased amount. Then multiply by 100.

Application 6.9

6.10 Finding Percentage Increase and Decrease

Activity

Study tip

Application 6.10

1. By what percentage will 900 be increased to become 1200?

2. When 2,000 is decreased by x% it becomes 1,600 Frw. Find the value of x.

3. The price of a shirt was increased by a percentage. The price increased from 6,000 to 8,000 Frw. Calculate the percentage increase.

4. The marked price of a car was 4,200,000 Frw. It was decreased by x% to become 4,000,000 Frw. What is the value of x?

5. The population of a country increased by a certain percentage. The population in the previous census was 12,000,000 people. The recent population rose to 14,400,000 people. By what percentage did the population increase?

6. The price of a pair of shoes decreased from 20,000 Frw to 17,000 Frw/ Find the percentage decrease?

6.11 Finding Percentage Profit and Percentage Loss

Activity

-Role-play buying and selling of items in a class shop.

- In pairs, demonstrate when you gain more money and when you get less money than the amount you purchased the item.

- Express the difference as a fraction of the buying price, then multiply by 100. How do you describe the result?

- Discuss your answers with the class.

Study tip

Application 6.11

1. Tereza sold a radio at 50,000 Frw. She had bought it at 40,000 Frw.Calculate the percentage profit.

2. Sales man made a loss after selling a dining table at 75,000 Frw which its original cost was 90,000 Frw. What was the percentage loss?

3. A grocer sold a pineapple at 500 Frw. She had purchased it at 400 Frw.Find the percentage profit.

4. Kwizera bought a school bag at 15,000 Frw. He realised that his books could not fit in it. He sold it at 12,000 Frw. What was the percentage loss?

5. A shopkeeper bought a sack of sugar at 750 Frw each kilogram. He sold every kilogram at 800 Frw. Calculate the percentage profit.

6. Mukandairo bought a dozen of counter books at 1,500 Frw each. She later sold every book at 1,000 Frw. What was her percentage loss?

6.12 Solving Problems Involving Percentages

Activity

- A school enrolment was 600 learners in 2016.

-The number increased by 20% in 2017.

- What is the enrollment in 2017?

- What is the meaning of 20% increase?

- Write a list of progressive increase of 20 to every 100 learners.

- What is the answer?

Study tip

- The original number is always a percentage (100%).

- Increase amount = (100% + % increase) of the original amount.

-Decrease amount = (100% - % decrease) of the original amount.

- A loss is a decrease and a profit is an increase.

Application 6.12

1. A cow increased its milk production from 15 litres by 10%. How much milk does it give now?

2. 450 people had accounts at a bank. 20% of the accounts were closed last month. How many people have accounts with the bank now?

3. The government tarmacked 150 roads last year. This year the number of tarmac roads has increased to 168. What is the percentage increase in the number of tarmac roads?

4. Catherine bought a dress at 5,000 Frw and sold it at 7,000 Frw. Calculate the percentage of profit.

5. A shopkeeper reduced the price of a shirt from 2,000 Frw to 1,500 Frw Calculate the percentage of decrease in the price of the shirt.

6. 300 kg of beans bought by the school was raised to 450 kg. Calculate the percentage of increase.

7. An article was bought at 50,000 Frw and sold at 46,000 Frw. What was the percentage of loss?

8. Mukandoli sold a sofa set at 400,000 Frw. She had bought it at 350,000 Frw. Find the percentage of profit.

6.13 Finding Ratios

Activity

In a farm, there are 3 goats to 5 sheep.

(a) Do you know how you can present the ratio of the goats to sheep ?

Write it down.

(d) Explain your working out.

Study tip

- A ratio is a comparison between quantities. No units are used.

- A ratio is an ordered pair of numbers written in the order a:b where b cannot be zero(0).

- Ratios may be used while calculating things which are compared proportionally.

Application 6.13

6.14 Sharing Quantities in Ratios

Activity

Get 32 bottle tops.

- In pairs, pick from the heap at intervals.

- When the 1st learner picks 3 bottles tops, the 2nd picks 5 until all the bottle tops are completely shared.

- How many bottle tops does each get?

- Share the procedure with other classmates.

Study tip

- Sharing in ratio; express each share as a fraction, then multiply by the quantity.

- Sharing in ratio; if the quantity of equivalent to some shares is given, get the unit quantity value to each share. Then multiply by the rest of the shares by the unit value.

- The sum of the given ratio is equal to the total quantity.

Application 6.14

6.14 Sharing Quantities in Ratios

Activity

Study tip

-Sharing in ratio; express each share as a fraction, then multiply by the quantity.

-Sharing in ratio; if the quantity of equivalent to some shares is given, get the unit quantity value to each share. Then multiply by the rest of the shares by the unit value.

- The sum of the given ratio is equal to the total quantity.

Application 6.14

1. Share 500 in the ratio of 2:3.

2. Share 420 kg in the ratio of 1:5.

3. Kaibanda and Mukamusoni shared 12,000 Frw in the ratio of 3:5. Find how much each got.

4. Two farmers shared 125 kg of beans to sow. How many kilograms did each get if they shared in a ratio of 3:2?

5. Anita and Alpha contributed money to buy land in the ratio of 7:8. Alpha contributed 640,000 Frw.

(a) How much more did Anita contribute? (b) Calculate the total amount of money both contributed.

6. The ratio of girls to boys in a primary school is 9:8. There are 450 girls in the school.

(a) Work out the number of boys. (b) Find the total enrolment in the school.

6.15 Increasing and Decreasing Quantities in Ratios

Activity

Get counters like beads or small stones.
Pick 30 of them and form groups of 5.
How many group have your formed?
Now form a 6th group of the same number of counters.
How many counters are in all the 6 groups altogether?
How many more counters have you added to the 30 counters?
Discuss and make a class presentation.

Study tip

Application 6.15

1. Increase 150 kg in the ratio of 7:3.

2. Decrease 490 in the ratio of 3:7.

3. Increase 720 in the ratio of 5:4.

4. Decrease 1,400 in the ratio of 7:10.

5. What amount do you get after increasing 21,000 Frw in a ratio of 9:7.

6. 800 books were decreased in the ratio of 3:4. How many books are there?

7. Increase 1,200 litres in the ratio of 5:8.

8. Decrease 4,000 Frw in the ratio of 5:8.

9. Find the amount you get after increasing 14,400 Frw in a ratio of 12:17.

6.16 Finding the Ratio of Increase and Decrease

Activity

Write 400 on slips of paper. Increase it to 600.
Express the new number as a fraction of the old number.
Reduce it to the simplest terms, then write as a ratio.
What do you notice?

Study tip

Ratio of increase = New amount to old amount. The new amount is bigger than the old amount.
Ratio of decrease = New amount to old amount. The new amount is smaller than the old amount.

Application 6.16

1. 500 was increased to 600. Find the ratio of increase.

2. 4,000 was reduced to 2,500. What is the ratio of decrease?

3. A trader used to purchase 144 cartons of markers. Now he purchases 168 cartons every week. Work out the ratio of increase.

4. The enrollment in a school dropped from 700 learners to 560 learners. What was the ratio of decrease?

5. What is the ratio increase from 108,000 to 156,000?

6. The price of a radio was 70,000 Frw. It was decreased to 63,000 Frw. Find the ratio of decrease.

7. The school fees in a school was 75,000 Frw. Now it is 90,000 Frw. What is the ratio of increase?

8. Mrs. Kwizera’s farm used to produce 900 litres of milk. It now produces 810 litres. Find the ratio of decrease.

6.17 Solving Problems Involving Ratios

Activity

Get 40 bottle tops.
Group them in twos and threes at the same time.
Write a statement about the above.
How many bottle tops are in each group?
Discuss, then make a presentation to class.

Study tip

Read and understand the question for clear interpretation.
Writing in ratio is a simple way of comparing number.

Application 6.17

1. The cost of a pen and a pencil is in the ratio of 5:2. What is the cost of the pen if the cost of a pencil is 100 Frw.

2. There are 120 sheep on a farm. The ratio of ewes to rams is 5:7. How many rams are in the farm?

3. The amount of money collected by P.6 learners to go for a picnic last year was 200,000 Frw. This year it increased in the ratio of 7:4. How much was collected this year?

4. The price of paraffin was 700 Frw a litre last month. It was increased to 900 Frw a litre. In what ratio was the price increased?

5. The marked price of a television is 124,000 Frw. A customer bought the television at a reduced price in a ratio of 3:4. How much d id she buy the television?

6. The population in Munyaneza’s district increased from 3,200 people to 4,800 people. In what ratio did the population increase?

7. Alpha and Anne got 360 saplings from the forestry official. They shared them in the ratio of 4:5 respectively.

(a) How many saplings did Alpha get?

(b) How many saplings did Anne get?

8. The ratio of boys to girls in a school is 9:11. If there are 330 girls;

(a) How many boys are in the school?

(b) Find the number of learners in the school.

6.18 Finding Indirect Proportions

Activity

Work done by 12 people can be completed in 5 hours.

If the time to accomplish the work is reduced, the number of people to accomplish the work should be increased.

(a) Discuss the situation.

(b) Why do you think the workers were increased as time reduced?

Study tip

When solving indirect proportion problems, you must note that:

Time to do any piece of work increases as the number of people decreases.
Time to do any piece of work decreases as the number of people increases.
A task to be completed in a short period of time needs many participants.
Work done by many people is shared, so it takes a shorter period of time to accomplish than work done by few people.

Application 6.18

6.19 Finding the Average Price of a Mixture

Activity

Take 3 kg of millet with 2 kg of maize. Ask how much 1 kg of millet and 1 kg of maize costs in the shop. Buy them and mix in a basket.

(a) Measure how many kilograms you have got altogether

(b) Calculate the money you have spent on your mixture.

(d) What do you notice?

(e) Why do you think it is important to study mixtures?

(f) Explain your working out to the class.

Study tip

To find the average price of the mixture:

First find the total cost of both types of mixture.
Then divide by the total kilograms of the mixture.

Application 6.19

1. Kambanda has 12 kg of brown sugar which cost 1,100 Frw per kilogram and 8 kg of white sugar which cost 1,200 Frw per kilogram. What will be the average price of the mixture? Show your working out.

2. Kalisa mixed 6 litres of honey which cost 4,500 Frw per litre with 9 litres of another quality of honey which cost 4,200 Frw per litre. Calculate the average price per litre of the mixture.

3. Charles mixed 30 litres of orange juice which cost 1,000 Frw per litre with 20 litres of passion fruit juice which cost 1,200 Frw per litre. Calculate the average price of each litre of juice. Show your working out.

4. Agnes mixed 20 kg of sorghum flour which cost, 2,000 Frw per kg with millet flour which cost 1,500 Frw per kg and formed a 40 kg mixture.

(a) Calculate the average price of the mixture. Show your working out.

(b) What was the mass of the millet flour?

6.20 Finding Quantity of One Type of the Mixture

Activity

-Get 2 kg of soya flour which costs 2,000 Frw per kilogram.

-Mix it with maize flour which costs 700 Frw per kilogram.

-Selling the mixture at 1,400 Frw, on which type of flour is a profit or loss achieved? And, by how much?

- What do you get?

- Selling maize flour alone, do you gain or make a loss? Explain.

- Try find the mass of the maize flour before forming a mixture. Make a class discussion.

Example

A mixture of yellow maize flour and white maize flour cost 800 Frw per kg.

4 kg of yellow maize flour cost 900 Frw per kg and the white maize flour

costs 700 Frw per kg. Find the kilograms for the white maize flo ur.

Show your working out.

Solution

1 kg of the mixture cost = 800 Frw

Selling the yellow maize flour alone causes a loss of:

(900 - 800) = 100 Frw per kilogram

Selling 4 kg results in a loss of:

(100 × 4) = 400 Frw

Selling the white maize flour alone results in a gain of

(800 - 700) = 100 Frw

The number of kilogram for the second type is ... 400 ÷ 100 = 4 kg

Study tip

To get a profit, the price of an item is lower than the price of the mixture.
To get a loss, the price of an item is higher than the price of the mixture.
Then balance it up by multiplying the ratio of profit and loss by the kilograms given.

To find the weight of the second item in a mixture, divide the total gain in the first item by the gain on each kilogram of the second item.

Application 6.20

1. Kamanzi mixed 5 kg of yellow beans which cost 600 Frw per kg with white beans which cost 700 Frw per kg. He sold the mixture at 600 Frw per kg. Find the number of kg for the white beans. Show your working.

2. Nkubiri mixed 10 kg of peas which cost 2,500 Frw per kg with another type which cost 1,200 Frw per kg and sold the mixture at 360 Frw. Find the number of kg for the second type of mixture. Show your working out.

3. Mutesi mixed 10 kg of white sugar that cost 1,300 Frw per kg with brown sugar that cost 1,100 Frw per kg. She sold the mixture at 1,200 Frw per kg. How many kg of brown sugar were there?

4. Kayinamura mixed 6 kg of yellow beans which cost 600 Frw per kg with coloured beans which cost 400 Frw per kg. If he sold the mixture at 500 Frw, how many kg of coloured beans did he mix?

5. Kaibanda mixed 5 kg of one type of rice which costs 2,000 per kg with another type which costs 1,600 Frw per kg. If he sold the mixture at 1,800 Frw, how many kg of another type did he mix?

6.21 Finding the Price of One Type of Ingredient in the Mixture

Activity

Kalisa mixed 20 kg of brown peas which cost 150 Frw per kg with 30 kg of green peas. Find the unit price of the green peas if the mixture cost 90 Frw per kg. Present your findings to the class.

Study tip

- After finding the total mass and the cost of the mixture, subtract the cost of the first type from the cost of the mixture. Lastly, divide the result by the weight of the second type.

Application 6.21

1. Jeanne mixed 40 kg of beans which cost 700 Frw per kg with 60 kg of another type. What is the price per kg of the second type if the price of the mixture is 600 Frw per kg? Show your working out.

2. Gasana has 120 kg of mixed sugar and sells each at 1,200 Frw. If there are 70 kg of white sugar which cost 1,300 Frw per kilogram, find the price per kilogram of the second type of sugar. Show your working out.

3. Geoffrey sold 150 kg of a mixture of Super rice and Pakistani rice at 1,600 Frw per kilogram. 65 kg was Super Rice at 2,000 Frw per kilogram. Calculate the price of Pakistani rice.

4. A farmer sells 50 kg of a mixture of groundnuts and simsim at 3,000 Frw per kilogram. She sold 20 kg of simsim at 4,000 Frw per kilogram. Find the price of the groundnuts.

5. Grace had 95 kg of mixed rice that she sold for 152,000 Frw altogether. She had bought 40 kg of one type of rice at 1,200 Frw per kilogram. Find the price of the second type of rice.

6.22 Finding both Quantities of a Mixture

Activity

Joan had 150 kg of mixed beans. The cost of the mixture was 650 Frw per kg. One type cost 700 Frw per kg and the second type cost 600 Frw per kg. What was the weight of each type of bean in Show your working out.

Study tip

To find the quantity of the two types in the mixture, use the ratio of each mixture over the total ratio and multiply by the cost of each mixture.

Application 6.22

1. In the shop, there are 60 kg of mixed beans. A kilogram costs 650 Frw. If the first type is sold at 700 Frw and another at 600 Frw, what is the weight of each type?

2. Mugabe mixed 120 kg of rice which he sells at 1,600 Frw per kilogram. The first type is sold at 2,000 Frw while the second is sold at 1,200 Frw. Find the quantity of each type.

3. Mutabazi mixed two types of sugar and formed 50 kg which he sold at 1,200 Frw per kg. If he sold the first type at 1,300 Frw per kilogram and the second type at 1,100 Frw per kilogram, find the kilograms of each type.

4. Umuhoza mixed two types of beans weighing 20 kg altogether and sold each kg at 900 Frw. The first type is sold at 1,200 Frw per kg and the second type at 700 Frw per kg. Find the kilograms of each type of beans.

5. Irish potatoes that cost 280 Frw per kg were mixed with another type of Irish potato that costs 240 Frw per kg to form a mixture. The price of 1 kg of mixed potatoes is 260 Frw per kg. Find the weight of each type of Irish potato if there were 60 kg of mixed Irish potato. Show your working out.

6.23 Solving Problems Involving Ratios, Percentages, Mixtures and

Inverse Proportions

Activity

Do you remember what you learned about ratios, percentages, mixtures and inverse proportions?

Why do you think it is important to study them?
If it is helpful to you, give an example for each and explain how they can be applied in real life situations.

Present your working out to the class.

Application 6.23

End of unit 6 assessment

1. The perimeter of a rectangle is equal to 280 metres. The ratio of its length to its width is 5:2. Find the area of a rectangle and show your working out.

2. There are (A) red flowers, (B) roses and (C) white flowers in a school garden. Write the ratio of the number of red flowers to the total number of flowers in terms of A, B and C.

3. If 35 men can reap a field in 8 days, in how many days can 20 men reap the same field?

4. Kambanda mixed 50 kg of sorghum flour which cost 2,000 Frw per kg with 20 kg of another type of sorghum. What is the price of a kg of the second type if the price of the mixture is 1,600 Frw per kg?

5. 6 boys working 5 hours a day can take 16 days to cultivate a garden. How many days will 4 boys take to do the same job, each working 5 hours?

6. There were 60 learners in a class. The number increased by 15%. What was the new number?

7. An examination paper has 25 questions. If Kamali got 70% of the answers correct, how many questions did Kamali fail? Show your working out.

8. How many people can dig 96 hectares in 8 days if 6 people can dig 64 hectares in 8 days?

9. In a boarding school of 600 learners, students have enough food for 90 days out of the whole term. How long will that food last if 50 more learners join the school?

10. How many kilograms of rice will 42 people need for 90 days if 10 people need 540 kilograms of rice for 72 days?

Key unit competence: To be able to convert between units of volume, capacity and mass.

Introduction

In daily life, people use different containers (bottles, jerry cans, buckets, etc to carry water or other liquids. Each container has volume, capacity and mass and when it is full of water it has a certain mass.

(a) Did you ever try to think about the relationship between the capacity of a bottle, the quantity of water to fill the bottle and the mass of the bottle full of water?

(b) Do you think that the volume of a container, its capacity and its mass may have a relationship? Explain.

7.1 Revision on Mass Measurement

Activity

- Take a stone outside of the classroom and measure its mass.

- Measure mass of different stones with different sizes and record the results in your note book.

- Explain the meaning of mass and give an example.

- What do you notice?

- Present your findings to the class.

Study tip

- The mass of an object is the amount of material in its weight.

- The standard unit of mass is kg (kilogram) and g (grams).

- For small quantities, g (gram) is used.

- For large quantities, use kilograms.

Application 7.1

7.2 Revision on Capacity Measurement

Remember that:
1 kl = 10 hl    1 kl = 100 dal 1 kl = 1,000 l
1 kl = 10,000 dl 1 kl = 100,000 cl    1 kl = 1,000,000 ml
1 hl = 10 dl    1 hl = 100 l 1 hl = 1,000 dl
1 hl = 10,000 cl    1 hl = 100,000 ml    1 dal = 10 l
1 dal = 100 dl 1 dal = 1,000 cl 1 dal = 100,000 ml
1 l = 10 dl 1 l = 100 cl    1 l = 1,000 ml
1 dl = 10 cl    1dl = 100 ml 1 dl = 10 ml

Activity

Get an empty jerrycan.
Fill it with water.
Pour water in one litre bottles.
How many litre bottles are filled from one jerrycan?
Compare the capacity of a jerrycan with 1 litre bottle.

Study tip

Capacity is the amount of liquid that fills a container.
The standard units of capacity are litres and milliliters: 1,000 ml = 1 l

Application 7.2

1. A tank of water contains 5 kl of water. How many jerrycans of 20 l can be fully filled from that tank?

2. A container of 10 l of juice was shared equally among 15 children, how many litres did each one get?

3. On a farm, 100 kl of milk is collected in a month. How many litres of milk is collected in 5 months if milk is produced at the same rate?

4. A jerrycan of 20 litres is poured into 3 litres small jerrycans.

(a) How many 3 litres small jerrycans are fully filled?

(b) How much water remains?

5. Convert 16.24 hl to dl.

6. How can you apply capacity measurement in your daily life?

7.3 Measurement of Volume

Activity

Find a box container, for example chalk box.
Measure the length, width and height of the box with a ruler.
Write down the results.
What is the volume of the box if:

(a) You measure in cm?

(b) You measure in mm?

List the units of volume.
How is the measurement of volume useful in daily life? Explain your

findings to the class.

Study tip

Volume is measured in mm3, cm3, dm3, m3, dam3.
Volume is the space occupied by an object.
Volume determines how big or how small an object is.
Conversion of units of volume is done better using a conversion table.

Application 7.3

7.4 Finding the Relationship between Units of Volume, Capacity and Mass

Activity

Find a rectangular cup or tin of one litre and fill it with wate r.
Measure the mass of water in the cup.
Measure and calculate the volume of the cup.
What is the relationship between these measurements?
Explain your working to the class.

Study tip

Volume, capacity and mass measurements are related only when measuring water.
1 l = 1 kg = 1 dm3.
This concept is helpful if you want to know the measurement of water when

you already know its measurement in one of the above measurements.

Application 7.4

1. Match the following units of measurements:

2. Find out how many litres are in a 20 kg bucket.

3. Does 1 litre of cooking oil equal 1 kg? Explain.

4. Why is the relationship between these measurements applicable when
measuring water?

7.5 Converting between Units of Volume, Capacity and Mass

Activity

Convert the following units:

10 kg = ...... kl = ...... cm3

1 dm3 = ...... ml = ...... dg

What do you notice?

Study tip

When converting capacity, volume and mass, use a conversion table including all the units.

Application 7.5

Key unit competence: To be able to calculate speed, distance and time, solve problems related to
different time zones and convert speed from km/hr to m/sec and vice versa.

Introduction

People move everyday, whether to school, home, place of work, visiting friends

and in any other journeys. How fast people move depends on the distance

and time. Imagine you are a business person who has to deliver goods to a

customer, who also needs to distribute them to his or her customers.

(a) What should you do? Travel faster or slower? By which means?

(b) What might be the result for late delivery of goods to the customer?

the distance traveled? Establish that relationship.

8.1 Comparing the 12-hour Format to the 24-hour Format

Activity

You have learned about telling time.
Choose any time in the morning and in the afternoon.
Tell time using the 12-hour and 24-hour formats.
Show time charts for 12-hour and 24-hour format.
Compare those formats and explain them.

Study tip

12-hour format starts immediately after midnight until 12.00 noon, then from 12:00 noon to 12:00 midnight.

24-hour clock starts immediately after 24:00 hours, then back to 24:00 hrs.
To change from 12-hour clock to 24-hour clock, add 12.00 hours to the given time if it is post meridiem or past mid day (p.m).

To change from 12-hour clock to 24-hour clock, add 00.00 hrs to the given time if it is ante meridiem or before mid day (a.m).

Application 8.1

8.2 Converting 12-hr to 24-hr Format

Activity

Do it personally, with your partner or in a group.

Match the time in 24-hour format with that in 12-hour format.

What do you notice?

Study tip

To change a.m. time to the 24-hr format, add 00:00 hours.
To change p.m. time to the 24-hr format, add 12:00 hours.

Application 8.2

Change the time from the 12-hour format to the 24-hour format.

(a) 4:21 p.m. (b) 5:56 p.m. (c) 9:12 a.m.

(d) 8:45 a.m. (e) 12:46 a.m. (f) 10: 43 a.m.

(g) 1:59 p.m. 7:18 a.m. (i) 3:14 a.m.

(j) 2:49 p.m. (k) 5:00 p.m. (l) 9:56 a.m.

8.3 Converting 24-hr Format to 12-hr Format

Activity

- A watch shows 22:22.

- It is in 24-hour format.

- What would it display in the 12-hours format?

- Explain to your classmates.

Study tip

To convert 24-hour format to the 12-hour format, time below 12:00 hours always add 00:00 hours.

To convert 24-hour format to 12-hour format, time above 12:00 hours always subtract 12:00 hours.

12:00 a.m. to 12:59 noon do not change. They remain as they are.
24-hours = 00:00 a.m. or 12:00 midnight, 00:40 hrs = 12:40 a.m, 12:30 p.m. = 12:30 hours.

Time above 12:00 hours, that is 12:00 hours to 24:00 hours, is always p.m.
Time below 12:00 hours, that is 00:01 hours to 11:59 is always a.m.
12-hours = 12:00 midday or 12:00 noon.

Application 8.3

Change the following time from 24-hour format to 12-hour format.

(a) 04:12 hr (b) 15:54 hr (c) 06:32 hr

(d) 22:10 hr (e) 01:23 hr (f) 23:45 hr

(g) 05:15 hr 17: 38 hr (i) 13:38 hr

(j) 03:45 hr (k) 19:40 hr (l) 09: 30 hr

8.4 The Concept of Time Zones

Activity

Have you ever felt you needed more time?
Have you ever wished you could turn the clock back?
Places that are West of your country’s time zone are earlier in time.
Places that are East of your country’s time zone are further in time.
Look at the world map showing standard time zones and answer the following:

(a) In how many time zones is the world divided?

(b) How many hours does each zone represent?

(d) What do you observe above time of places left of Rwanda?

(e) What about time of countries to Rwanda’s right?

Study tip

A time zone is a region of the globe that observes a uniform standard time for legal, commercial and social purposes.

The world is divided into 24 time zones.
Each time zone represents 1 hour difference in time.
Moving to the West we reduce time and to the East, we add time. One hour per time zone.

Remember that between two longitudes (time zones) there is 15O which is equal to 1 hour.

Application 8.4

Use the world map showing standard time zones to answer the following:

Find the time reading in the time zones below:

(a) 2nd time zone to the East. (b) 7th time zone to the West.

(e) 9th time zone to the East (f) 5th time zone to the West.

(g) Time at 0 time zone. 12th time zone to the East.

8.5 Solving Mathematical Problems Relating to Time Zones

Activity

Study the map of the world showing the standard time zones.
Name the meridian that is representing 0.
How important is the meridian that you named?
What important line is represented by -12 or +12?
Give the importance of that line.

Study tip

To find time of another time zone to the West, multiply its position from the

time in the given time zone by 1 hour. Then subtract from the given time.

To find time of another time zone to the East, multiply its position from the

time in the given time zone by 1 hour. Then add to the given time.

In all calculations, use the 24-hour format to make calculations easier.
There is a difference of one hour between two time zones.

Application 8.5

8.6 Calculating Speed

Activity

Take a stop clock.

Start it and start running around the field once.
Stop it when you return to the starting point.
Count the time you have taken.
Divide the distance with the time you took.
What do you notice?

Study tip

Application 8.6

Work out the following:

1. A bus travelled for 3 hours to cover a distance of 180 km. At what speed was the bus moving?

2. At what speed was the car travelling if it covered 540 km in 6 hours?

3. What is the speed, when the distance and time are:

(a) 160 km and 4 hours?

(b) 150 km and 3 hours?

4. A trailer left city A for city B covering 720 km. If it took 12-hours to arrive, what was its speed?

5. A truck covered a distance of 3,600 metres in 3 minutes. Calculate its speed in metres per second.

8.7 Converting Speed from km/hr to m/sec

Activity

Muziranenge ran 6 km in one hour.

- If she was running at the same rate, can you find the distance in metres she covered in one second?

- Discuss and defend your answer by showing your working steps?

Study tip

Application 8.7

Work out the following:

1. Change the following km/hr into m/sec:

(a) 90 km/hr (b) 60 km/hr (c) 180 km/hr

(d) 54 km/hr (e) 252 km/hr

2. The distance from village A to village B is 720 km. A car takes 6 hours

to cover the journey. Calculate its speed in m/sec.

3. Jane covered a distance of 50 km in 2 hours.

(a) Calculate the speed in km/hr.

(d) What was the speed in m/sec?

4. The distance from one town to another town is 216 km. Kalisa covered the distance in 4 hours. Calculate the speed in m/sec.

5. A truck driver covered a distance at a speed of 70 km/hr. Express the speed in m/sec.

8.8 Converting Speed from m/sec to km/hr

Activity

During breaktime, learners of P.6 had a 100 m race.

- The winner covered 5 metres in one second.

- Suppose he runs at the same rate, what distance
can he cover in one hour?

- Explain your working out.

Study tip

Application 8.8

Work out the following

1. Convert the following into km/hr:

(a) 15 m/sec (b) 45 m/sec (c) 25 m/sec (d) 100 m/sec

2. An aeroplane covered 1,000 metres in 5 seconds. Express the speed in km/hr.

3. During a bicycle Rwanda Tour Racing, a cyclist covered 100 metres in 10 seconds. Calculate the speed in kilometres per hour.

4. In a school competition, one athlete ran 100 metres in 15 seconds. What was the speed in kilometres per hour?

5. A cyclist was riding at a speed of 10 mertes every second. Change his speed into kilometres per hour.

6. In the East African Safari Rally, Mansoor drove at a speed of 40m/sec. What speed was he driving in kilometres per hour?

8.9 Calculating distance

Activity

Mugenzi and Mutoni had a journey from home to visit a friend called James.
They took 2 hours moving at a speed of 13 kilometres per hour to reach there.
What is the distance from their home to James’ home?
Explain your working out to the class.

Study tip

Distance = speed × time.
If departure and arrival time is given, calculate the duration to get the time.
Distance is measured in kilometres (km), hectometres (hm), decimetres (dam), metres (m), centimetres (cm) and millimetres (mm).

The standard unit of measuring distance is metre (m).

Application 8.9

8.10 Calculating Time

Activity

Joseph walks to school every day.
The distance from home to school is 3 km.
One day, he walked at a speed of 2km/hr. How long did he take to reach school?
Present your working out to the class.

Study tip

Application 8.10

Work out the following:

1. Christine drives her car at a speed of 72 km/hr. What time does she take for the whole journey if she drives to cover a distance of 216 km?

2. A bus travelled from Kigali to Kampala. It was moving at a speed of 60 km/h. It stopped after moving 180 km. What time did it take to cover the journey?

3. Murengezi walks at 2 km/hr when he is going to school. If he moves a distance of 3 km, how long does it take him to arrive at school?

4. How long did a car travelling at 60 km/h take to cover 240 km?

5. How long will it take a cyclist to cover a distance of 105 km at a speed of 35 km/hr?

6. A bus covered a distance of 455 km at a speed of 79 km/hr. Calculate the time it took to arrive at the destination.

8.11 Moving Bodies towards Each Other

Activity

You studied how to calculate speed, distance and time.
Try to solve the following:

Mugenzi drove a car from city (A) towards city B at 7:00 a.m. He was travelling at a speed of 50 km/hr and Mukamukunzi left city (B) at the same time driving towards city A at the speed of 40 km/hr on the same road. If its is 100 km from city (A) to city (B):

(a) At what time did they meet?

(b) What is the distance each had covered by the time they met?

Explain your working out.
Present your work to the class.

Study tip

Distance is worked out by finding the product of speed and time.
Duration is the time between two events.
Duration is worked out by subtracting the starting time (ST) from the ending time (ET).

To get distance covered by each, when they meet, multiply each one’s speed by the duration.

Application 8.11

Work out the following:

1. Two motorcyclists started the journey at 9:00 a.m. and met at 11:00 a.m. One of them moved from town (A) at a speed of 45 km/hr and another one moved from town (B) at a speed of 35 km/hr. What distance had each covered by the time they met?

2. Clementine and Justin were at a distance of 60 km apart. They left their homes at 7:00 a.m and met at 9:00 p.m. If Justin moved at a speed of 12 km/h, what was the speed of Clementine?

3. Tom and Silas started the journey towards each other; moving at 30 km/hr and 24 km/hr respectively. If between their cities there are 135 km. At what time did they meet?

4. Mugenzi and Murengezi left their residences and moved towards each other at 9:00 a.m. They met at 11:40 a.m. Their speeds were 40 km/hr and 35 km/hr respectively. What was the distance between them?

5. Town A and Town B are 240 km apart. Anna started travelling from town A at the same time, Chantal moved from town B. Chantal was moving at a speed of 50 km/hr. They met after 3 hours. Find Anna’s speed.

8.12 Moving Bodies Following Each Other

Activity

A car left Musanze at 8:00 a.m moving at a speed of 40 km/hr and the second car also left Musanze at 9:00 a.m moving at a speed of 60 km/hr.

(a) When do you think the second car will catch up with the first car?

(b) What distance will they have covered?

Example
A bus travelling at 40 km/hr left Kigali at 8:30 a.m. Another bus travelling at 60 km/hr followed it after 1 hour. When did the second bus over take the first bus? Show your working.

Solution
The first bus was covering 40 km in 1 hour.
The first bus had covered 40 km/hr before the second started.
Difference in speed = (Second speed - first speed)
= (60 km/hr - 40 km/hr)
= 20 km/hr

Study tip

Distance is a product of speed and time.
To work out the distance covered by one object following another, find the difference between the speeds.

To find the duration the second moving object overtakes the first, divide the speed of the first object by the difference in distance moved by the two objects after 1 hour.

To find the time the second moving object overtakes the first moving object, add the duration for overtaking to 1 hour.

Application 8.12

Work out the following:

1. A car moving at a speed of 60 km/hr followed a bus which had departed 2 hours earlier. If the bus was traveling at 45 km/hr, when did the car overtake the bus if the bus left at 9:00 a.m?

2. Car A moves at a speed of 40 km/hr and car B moves at a speed of 60 km/hr. What is the distance between them after 3 hours if they are moving in the same direction?

3. Mukakalisa left Kayonza at 8:00 a.m moving at 60 km/hr and Muhire followed her and met her at 11:00 a.m. What distance had Muhire covered?

4. A cyclist moving at 30 km/hr started from city (A). After 2 hours a lorry followed, moving at 40 km/hr.

(a) How long did it take the lorry to catch up with the cyclist?

(b) What distance had both covered?

8.13: Calculating Average Speed

Activity

Study tip

Application 8.13

End of unit 8 assessment

1. Angel and Karemera started moving towards each other at 7:00 a.m. Their speeds were 20 km/hr and 30 km/hr respectively. The distance between them was 150 km.

(a) Find the time they met.

(b) Find the distance each had covered.

2. A car moving at a speed of 70 km/hr and a lorry moving at a speed of 30 km/hr started moving towards each other at 8:00 a.m. What was the distance between them if they met at 10:30 a.m?

3. The distance from town A to town B is 720 km. A car takes 8 hours to cover the journey. Calculate its speed in m/sec.

4. Johnson is in Bangladesh which is at 6 time zones East. He wants to be in Greenland which is 2 time zones West at 16:30 hr. What time should he fly?

5. Two motorists started moving at the same time towards each other. One motorist was moving at 60 km/hr and another at 80 km/hr. They met after 4 hours.

(a) What distance did both cover after 1 hour?

(b) What was the distance between them before they started moving?

6. A lorry and a bus were travelling in the same direction. They set off at the same time from the park. The lorry was travelling at 50 km/hr. The bus was travelling at 70 km/hr.

(a) How far apart will both be after 3 hours?

(b) What will be their a part distance after 5 hours?

7. A bus takes 4 hours to cover a journey of 240 km and takes only 2 hours to return. Calculate the average speed.

8. A truck took 3 hours to cover a journey at 4512 km/hr. On return journey, it took 4 hours. Find its average speed.

9. It is 8:20 pm in a city. What time is it in the 5th time zone to the East?

10. Lucimango arrived at 9:15 p.m. to his destination. He started his journey from the 7th time zone East of his destination. What time did he start the journey?

Key unit competence:

To be able to work out simple interest and solve problems involving saving

Introduction

In business, people aim at getting more money in order to become rich. Many people become rich by investing their money in businesses. When people bank, borrow or lend money, they expect to get more added money to their originally used money and this is called interest. Consider the following situation and answer the related questions: A person put money in the bank for a year amounting to 200,000 Frw. The bank gave him/her 10 out of each 100 Frw for using his/her money.

(a) How much more money was he/she given? (b) Why do you think we should save our money in the bank?

(c) Can you give different possibilities of using money and getting interest without saving it in the bank?

(d) How does interest benefit us in daily life?

9.1 Calculating Simple Interest

Activity

Visit a bank or SACCO if available.

Ask how it operates.
Write down the information on a sheet of paper.
Back to class, make a class discussion

example

Study tip

The money borrowed, saved or lent is Principle (P).
The percentage used to calculate interest is Rate (R).
Rate is the amount on every one hundred of the Principle that is earned or paid back.
The period in years that the principle is invested is Time (T).
The additional amount offered or paid is Simple Interest (S.I).
To find simple interest (S.I), use S.I = P x R/100 x T.

Unit 9: Simple Interest and Problems Involving Saving

Application 9.1

Calculate simple interest (S.I) given;
1. Principle = 400,000 Frw, Rate = 5% p.a, Time = 1 year.
2. Principle = 650,000 Frw, Rate = 12% p.a, Time = 3 years.
3. Principle = 800,000 Frw, Rate = 15% p.a, Time = 4 years.
4. Principle = 1,000,000 Frw, Rate = 18% p.a, Time = 2 years.
5. Principle = 1,200,000 Frw, Rate = 20% p.a, Time = 5 years.
6. Principle = 1,350,000 Frw, Rate = 24% p.a, Time = 3 years.

9.2 More about Calculating Simple Interest

Activity

example

Study tip

If rate is a fractional percentage, change it into a common fraction first

If time is in months, change it into years first.
Remember 12 months make one year. Express the months given out of 12.

Application 9.2

9.3 Solving Problems Involving Simple Interest

Activity

Suppose you deposit 6,500,000 Frw in a bank.
The bank promises to give you interest rate of 10% per year in 3 years.
What will be your simple interest after the 3 years?
Explain how you get it and then present your working out to the class.

Study tip

Read and interpret the words in the problem correctly.
Identify the principle, rate and time.
The money banked borrowed or lent is Principle (P).
The percentage used to calculate interest is Rate (R).
The period that the principle is invested is the Time (T).
The additional amount offered or paid back is the simple interest (S.I).
Simple interest = P × R/100 × T.

Application 9.3

work

9.4 Calculating Interest Rate

Activity

the formula

Study tip

Application 9.4

9.5 Solving Problems Involving Interest Rate

Activity

Your father has a loan of 800,000 Frw from a bank.
In one year, he pays interest of 96,000 Frw.
Help him to calculate the interest rate.
Present your working out as you explain to him how to get it.

Study tip

Application 9.5

9.6 Calculating Principle

Activity

On a slip of paper, write S.I formula
Multiply both sides by 100.
Divide both sides by R.
Then divide both sides by T.
Write the expression as an equation equated to P.
Now attempt this:
Write 20,000 Frw on a slip of paper. Multiply it by 100.
Divide the product by 10, then by 4.
What is your result?
Present your working out to the class.

example

Study tip

Application 9.6

Find the principle given:
1. Simple interest is 2,000 Frw, rate is 2% p.a and time is 2 years.
2. Simple interest is 45,000 Frw, rate is 3% and time is 9 months.
3. Rate is 4% p.a, time is 3 years and simple interest is 60,000 Frw.
4. Time is 5 years, simple interest is 72,000 Frw and rate is 20% p.a.
5. Simple interest is 15,000 Frw, rate is 3% p.a. and time is 2 years.
6. Rate is 12.5% p.a., simple interest is 10,000 Frw and time is 3 months.
7. Time is 6 months, rate is 60% p.a and simple interest is 24,600.
8. Simple interest is 14,400 Frw, rate is 12% per year and time is 12 months.

9.7 Solving Problems Involving Principle

Activity

Alpha deposited some money in the bank. If at the rate of 4% in 2 years
he earned a simple interest of 5,000 Frw, what is the money he banked?

Explain how you get it.

examples

Study tip

Study tip

Application 9.7

Work out the following:
1. Mary banked money at 5% simple interest rate and earned an interest
of 5,000 Frw in 5 years. How much did she bank?
2. Kambabazi borrowed money that earned her 45,600 Frw as interest
in 4 years at a simple interest rate of 12%. Calculate the amount she
borrowed.
3. Mutangana banked money in a bank that gives an interest rate of 10%
per year. 4 years later, he earned an interest of 65,000 Frw. What was
the principle?
4. Twagirimana lent money and earned an interest of 64,000 Frw at a rate
of 10% per year in 2 years. How much did he lend out?
5. Kalisa borrowed money. He wanted to boost his grocery. After 6 months,
he paid 87,000 Frw at a rate of 12% per year. How much did he borrow?
6. Kanyange banked money in Equity Bank. She was offered a simple
interest rate of 24% per year. She withdrew it after 6 months. Calculate
the amount she banked if she was given an interest of 96,000 Frw.
7. Mukandairo save money in a SACCO. She was offered a simple interest
rate of 20% per year. How much did she save if she was given 160,000
Frw as interest 4 years later?

9.8 Calculating Time

Activity

Get a slip of paper.
Multiply simple interest by 100 both sides.

Now divide the product by the product of P and R both sides.
What expression is formed?
Equate T to the expression to form an equation.
Present your result to the class.

Study tip

Application 9.8

Calculate time given:
1. Simple Interest is 20,000 Frw, Principle is 200,000 Frw and interest rate is
5% per year.
2. Principle is 100,000 Frw, simple interest is 10,000 Frw and interest rate
is 10% p.a.
3. Interest rate is 5% p.a, principle is 1,200,000 Frw and simple interest is
40,000 Frw.
4. Simple interest is 20,000 Frw, interest rate is 10%, principle is 400,000 Frw.
5. Interest rate is 25%, simple interest is 1,000,000 Frw and principle is
4,000,000 Frw.

6. Principle is 6,000,000 Frw, interest rate is 4% p.a. and simple interest
is 240,000 Frw.
7. Simple interest is 144,000 Frw, Principle is 720,000 Frw rate is 2% p.a.
8. Interest rate is 10% p.a. principle is 2,480,000 Frw and S.I is 124,000 Frw

9.9 Solving Problems Involving Time

Activity

How long do you take to earn 4,000 Frw as simple interest at the rate of 4% p.a.
when you deposit 50,000 Frw in a bank?
Explain your working out to the class.
Example 1 Example

Study tip

Application 9.9

Work out the following:
1. Kayitesi deposited 18,000 Frw in a bank that gives an interest rate of 10%
per year. How long did it take her to get 1,800 Frw as simple interest?
2. Mr Ntwari borrowed 50,000 Frw from a bank that gives an interest rate
of 10% p.a. How long did she use the money if she earned an interest
of 25,000 Frw?
3. Nsenga got an interest of 70,000 Frw at a rate of 20% per year. How
long did it take him if he deposited 350,000 Frw?
4. Munyandekwe lent Mukandori 240,000 Frw at an interest rate of 5% per
year. After how long will he get interest of 30,000 Frw?
5. How long will 720,000 Frw take to amount to 960,000 Frw at a rate of 5%?
6. Muhakanizi deposited 1,000,000 Frw in KCB at an interest rate of 12%
p.a. He earned 240,000 Frw as interest. How long was the money in
the bank?

9.10 Calculating Amount

Activity

Take a slip of paper.
Write principle, rate and time values of your choice.
Now find the sum of interest and principle.
What is your answer?
Explain your working out to your classmates.

example

Study tip

Amount is the sum of principle and interest.

To find amount, first calculate interest, then add it to principle.

Application 9.10

find the amount

9.11 Solving Problems Involving Amount

Activity

Suppose you are a manager of certain Bank.

Role-play a situation of a person who came to apply for a loan in your Bank.
Show how the loan is processed and then identify what the manager

does in order to help the bank and the client to achieve their business
objectives.

Explain your point of view.
Present your work.

example

Study tip

Amount is the total sum of money in the bank after the principle and the
interest earned over a given period.

Amount = Principle + interest earned.

Application 9.11

kamana

9.12 Saving Money in the Bank or Putting it in Investments

Activity

You are aware that successful people, institutions and businesspeople
save money in the banks or in investments.

Suppose you are given 20,000 Frw, write down different ways of saving from it.
Discuss and give an example of the following:

(i) the advantage of saving money in banks.
(ii) the advantages of putting money in investments.

Present your working out to the class.

Example 1

Suppose you are given 10,000 Frw. Make a budget and show how you can
save from it.
Solution
Budget: one dozen of books for 3,000 Frw, one graph book for 500 Frw, one
handkerchief for 500 Frw, six pens 1,000 Frw, oneMaths set 2,000 Frw
Expenditure = (3,000 + 500 + 500 + 1,000 + 2,000) Frw = 7,000 Frw
Expenditure = 7,000 Frw
Saving = Owned money – Expenditure
= (10,000 – 7,000) Frw
= 3,000 Frw

example

Study tip

Savings is the amount of money kept after spending some of it on basic needs.
Savings is calculated by subtracting expenditure from the money owned
Savings = Income - Expenditure
Banks or investments give additional money to the saver or investor. It is because they use the saved money to earn more money.

Application 9.12

work

find

9.13 Solving Problems Involving Savings

Activity

John wants to get an interest of 3,000 Frw in one year. How much must
he invest for that period at 8%?
Explain your working step.

Example 1

Gabriel saved money in a bank. He was offered 1,500 Frw monthly as interest.
If he deposited 1,350,000 Frw. How much did he have on his account after 2
years?
Solution
1 year = 12 months
2 years = 2 x 12 = 24 months
In 1 month he got interest of 1,500 Frw
In 24 months he got interest of (1,500 x 12) = 180,000 Frw
Amount = Principle + Interest
= 1,350,000 + 180,000
= 1,530,000 Frw.
Gabriel had 1,530,000 Frw after two years.

Example 2

Kayitesi has been paid 4,000 Frw by one of her business partners. She wants
to save this money.
(a) Advise her on the different methods of saving.
(b) Portray the importance of saving to Kayitesi.
Solution
(a) - She can deposit the money in a bank to earn interest.
- She can invest the money by buying shares in different investments
such as companies, co-operative societies, and others.
- If it is the only amount she has, she should spend some of it on needs,
but not all. She must save some in the bank or SACCO.
(b) - Kayitesi earns more money in form of interest.
- She will improve her standard of living by earning more money in future.
- She will always be having money available to sustain her needs.

Study tip

Savings is the amount of money kept after spending some of it on basic needs.
Savings is calculated by subtracting expenditure from the money owned.
Savings = Income - Expenditure.
Banks or investments give additional money to the saver or investor. It isbecause they use the saved money to earn more money.

Application 9.13

work

End of unit 9 assessment

1. Find simple interest earned on 150,000Frw at an interest rate of 30% per year.

2. Mushumba earned an interest of 4,500Frw for 2 year at 5% interest rate

per year. What was the Principle?

3. How long will 700,000Frw take to gain an interest of 140,000Frw at an

interest rate of 5%?

4. Calculate the simple interest on 25,000Frw for 2 years at an interest rate

of 10% per year.

5. How long will Gasana earn 24,000Frw if she deposited 120,000Frw in a

bank which offer her an interest rate of 10% per year?

6. Find principle that will earn 10,000Frw for Gihozo. The interest rate offered

is 5%7. Nkundimana invested 8,406,500Frw. He realised a total amount of 9,026,300Frw. How much interest did he gain?

8. Dusabe borrowed 1,200,000Frw from a bank and invested it for 8 months.

The interest rate offered was 612

% per year. How much did he pay altogether?

9. 152,400Frw was gained from 286,000Frw by Moses. He had kept his

money in a SACCO for 8 months. Calculate the interest rate.

10. Show how you would spend 25,000Frw and remain with savings.

11. Umurisa invested money in a SACCO for a period of time. She had 9,600,000Frw on her account. She was given 1,052,000Frw as interest. How much

did she invest?

12. 7,200,000Frw was invested by a co-operative society. It was offered an

interest rate of 1219% per year. The time for investing the money was

9months. Calculate the total amount on the account.

13. Kamanzi deposited 1,280,000Frw in a bank for 8 months. He realised an

interest of 64,000Frw. Calculate the interest rate the bank offered him.

14. Lucumu invested 8,400,000Frw in a company. He was offered interest

of 20,000Frw.

(a) How much interest had he after 2 years?

(b) How much money did he have on the account after 314

years?

15. Irebe borrowed 480,000 from a SACCO for 112

years. The interest rate offered was 1212per year.

(a) How much interest did she pay back?

(b) What amount did she pay the SACCO? for 6 months.

Key unit competence: To be able to write sequences of whole numbers, fractions and decimals.

Introduction
Pour 3 litres of water in a small jerrycan and mark the point. Remove the water and then get a soda bottle of 300 ml. Put the water in a bottle at a time as you pour in the same jerrycan until you fill it to the same level. What do you observe?
(a) Find out the number of soda bottles to fill the jerrycan at the same level as for 3litre bottles.
(b) After your experience, are 3 litters of water equivalent to 10 times the water in the bottle of soda containing 300 ml? Justify your answer.

10.1 Algebraic Expressions

Activity

Write the following algebraic expressions.
1. Twice a number added to 4.
2. The product of 2 and a number minus 3.

3. The difference between a number and 5.
4. A number multiplied by 3 take away 1

Study tip

Correctly interpret the operation terms used.
The terms which relate to (+) are; add, sum, plus, more and increase.
The terms which relate to (-) are; subtract, reduce, difference, take away,
decrease, and less.
The terms which relate to (x) are; multiply, product and times.
The terms which relate to (÷) are; divide, share, average and distribute.

Application 10.1

Write the following algebraic expressions for these:

1. 3 times the difference between p and q.

2. Divide thrice the product of m and n by 7.

3. Divide the sum of y and 6 by 5.

4. Twice x take away 6, then multiply the result by 3.

5. Half of the product of x and y.

6. The sum of p and q multiplied by the product of r and s.

10.2 Equivalent Expressions

Activity

Collect 6 pens and 4 rubbers.
Form 2 groups of each item.
Write an algebraic expression relating the collected items to the grouped ones.

Study tip

To find equivalent expressions, open brackets then simplify.
The expression on the left has to be equivalent to the expression on the right.

Application 10.2

10.3 Finding the Missing Consecutive Numbers

Activity

Study tip

To get the next number in a linear sequence, work out the common difference.
Then continue adding the common difference to find the missing numbers.

Application 10.3

Find the missing numbers in the sequences below;
(a) 3, 8, 13, 18, , , , ,.
(b) 1, 4, 7, 10, , , , ,.
(c) 2, 5, 8, 11, , , , ,.
(d) 5, 9, 13, 17, , , , ,.
(e) 1, 7, 13, 19, , , , ,.
(f) 3, 14, 25, 36, , , , ,.
(f) 4, 10, 16, 22, , , , .
(f) 9, 22, 35, 48, , , , ,.

10.4 Finding the Missing Consecutive Fractions and Decimals

Activity

Study tip

To get the next fraction or decimal in a linear sequence, work out the common difference.

Then continue adding the common difference to find the missing fraction or decimal.

Application 10.4

10.5 Finding the General Term/rule of a Linear Sequence

Activity

Study tip

To find the general rule for a sequence, multiply the positive by the common difference, then add or subtract a constant.

Application 10.5

Find the general term/rule for the linear sequences below.
(a) 5, 9, 13, 17, , ,    (b) 2, 4, 6, 8, , ,
(c) 3, 11, 19, 27, 35, , , (d) 7, 12, 17, 22, 27, , ,
(e) 6, 10, 14, 18, , ,    (f) 3, 6, 9, 12, , ,
(g) 1, 4, 7, 10, , ,    3, 8, 13, 18, , ,
(i) 2, 8, 14, 20, , ,    (j) 1, 3, 5, 7, , ,
(k) 2, 5, 8, 11, , , (l) 1, 5, 9, 13, , ,

10.6 Finding the General Term/rule of Linear Sequence for Fractions and Decimals

Activity

Study tip

First find the common difference, then multiply it by the order of term.
If the product is not equal to the term, add or subtract a constant.

Application 10.6

10.7 Finding the Missing Number or nth Term in a Linear Sequence

Activity

Find the general rule for the linear sequence: 1, 3, 5, 7, , ,
Now, taking n to be the order of term, calculate to find the 12th term.
Explain to the class.

Study tip

To find the nth term in a given sequence, first find the general rule.
Multiply the position by the common difference and decide on how to get each term including the nth term.

Use the general rule to find the nth term, by substitution.

Application 10.7

Find the missing number
(a) Find the 15th term in the sequence: 5, 8, 11, 14, 17, , , , .
(b) Find the 19th term in the sequence: 1, 3, 5, 7, 9, , , , .
(c) Find the 20th term in the sequence: 2, 7, 12, 17, 22, , , , .
(d) What is the 30th term in the sequence: 3, 7, 11, 15, 19, , , , .
(e) What is the 26th term in the sequence: 2, 4, 6, 8, , , , .
(f) Find the 40th term in the sequence: 2, 9, 16, 23, 30, , , , .
(g) What is the 99th term in the sequence: 4, 7, 10, 13, 16, , , , .
Find the 100th term in the sequence: 7, 13, 19, 25, , , , .

10.8 Finding the Missing Fraction or nth Term in a Linear Sequence

Activity

Study tip

Find the general rule for the linear sequence.
Substitute the value of nth term for n, then solve the equation to get the nth term.

Application 10.8

10.9 Finding the Number Sequence Using the General Term/rule

Activity

Study tip

To find the linear sequence, substitute the order of term for n in the

general term/rule. Then solve.

Application 10.9

End of Unit 10 Assessment

1. Write the following algebraic expression:

(a) Subtract 6 from n, then multiply by 2.

(b) The sum of m and n divided by 4.

(d) A quarter the product of a and b.

2. Find the equivalent expressions.

(a) 3(2x – 4) = 2(3x – 6)

(b)12 (4y + 2) = 2y + 1

(d) (x + 3) + 2(x – 1) = 3x + 2

(e) 169m – 13n = 13(13m – n)

(f)1/4 (28x – 8) =1/2 (14x – 4)

3. Find the missing consecutive numbers.

(a) 1, 5, 9, 13, , , , .

(b) 3, 7, 11, 15, , , , .

(d) 3, 6, 9, 12, , , , .

(e) 1, 6, 11, 16, , , , .

(f) 2, 6, 10, 14, , , , .

4. Find the general rule for the linear sequences below;

(a) 1, 4, 7, 10, 13, , , , .

(b) 2, 6, 10, 14, , , , .

(d) 7, 16, 25, 34, , , , .

Key unit competence: To be able to form and solve simple algebraic equations and inequalities.

Introduction

In daily life situations, people are faced mathematical problems that require simple and complex calculations. In most cases, different techniques are used to solve those problems. Some people prefer to use only numbers (arithmetic methods) while others prefer to use numbers mixed with variables/letters (algebraic methods).
Consider the following situation and answer the related questions:
In a certain classroom, x represents the number of boys while the number of girls is 2 times the number of boys. The total number of students is 30.
(a) How do you think the above mathematical problem can be solved? Using arithmetic method or algebraic method?
(b) Try to change the above mathematical problem into algebraic equation and find out the answer.

11.1 Like Terms of Algebraic expressions

Activity

Write this expression on paper slips: 7x + 6x + 3a + a + 11x + 2a

Group the same terms together.
Then add them.
What do you notice?
Give your observation to the class.

Study tip
Like terms have similar items or expressions.
First collect like terms then simplify the expressions.
Collecting like terms involves application of integers.

Application 11.1

Simplify these expressions:
(a) 4x + 5x =
(b) 2y + 4y - 2y =
(c) 2p + 4q + 3p - 2p =
(d) 9mno - 6mno + 3xy - 2ab +3xy + 6ab =
(e) 5xyz - 3xyz + 2xzy =
(f) A rectangle has length 2a - d and width 4a - 3d. Find its perimeter.
(g) A square has sides of 3x + 6m. Find its perimeter.

11.2 Unlike Terms of Algebraic Expressions

Activity

-There are 40 cattle (c), 16 sheep (s) and 32 goats (g) on a farm.
- Write down an algebraic expression for the animals above.
- Is there any way of grouping same animals together?
- Discuss, then make a presentation to class.

Study tip

Unlike terms have expressions or items which are not similar.
When simplifying algebraic expressions, unlike terms are not simplified because

they do not have similar terms or expressions that can be collected together.

Unlike terms differ letters and powers but coefficients may be the same.

Application 11.2

Simplify the following:
(a) 3xyz – 3xyr + 4 xyz + abc (b) 12x2y – 4xy2 – 5a – 19b – a
(c) 4x – xz – x + 6yx + 4tz (d) 17abfg – 3abcd – 5acd – 10defg
(e) 5abc – 4bca + 2acd – abc (f) fgh – fjk + 3fkj + 2fjk
(g) 10pqr + 9prq – 4rxy – 6pqr stu – 4tuv + 7rst + 2tuv

11.3 Substituting Algebraic Expressions with Addition and Subtraction

Activity

If x = 5 and y = 4, evaluate: x + 6y
If w = 5, x = 3 and y = 6, find the value of wxy.
Explain your answers to the class.

Study tip

To substitute is to replace. To work out algebraic expressions, replace the letters with the given numbers and simplify.

Application 11.3

Work out the following:
If x = 10, y = 2 and t = 5, p = 6, q = 8, r = 4; evaluate the following:
(a) x – 3 (b) t + y (c) y + q – t
(d) q + r – p (e) p + x + r – t (f) p + q + y
(g) x + y + t q + r – y (i) x + r + p
(j) q + p – r (k) t + x + p – r (l) y + p + t
(m) x + r + p y + r – p (o) x + r – p

11.4 Substituting Algebraic Expressions Involving Multiplication

Activity

Write 4 letters and equate them to values of your choice.
Form multiplication statements for two or more letters.
Substitute their values, then work out the products.
Explain your working out to the class.

Study tip

To work out algebraic expressions with multiplication, expand then substitute the values.

Simply to obtain the answer.

Application 11.4

11.5 Substituting Algebraic Expressions Involving Division

Activity

Study the cards below:

-Select numbers of your choice, substitute and work out.
-Share your working out with classmates.

Study tip

Dividing algebraic expressions involves applying the concept of dividing integers.
To work out substituting algebraic expressions involving division, substitute into the equation, simply then divide.

Application 11.5

11.6 Simple Algebraic Equations with One Unknown

Activity

An equation consists two parts which are equal to one another as shown below.
-Get a beam balance.
- Place small stones on one side and some fruits like oranges or mangoes on the other.
-Do they balance?
- If not, remove some items on the heavier side or add some items on the lighter side to balance the two sides.
- Now, given x + 2 = 18, what number must be put in place of x so that the sides are equal?
- Explain how you get the number.
- Present your working out to the class.

Study tip

To solve an equation involving brackets, remove brackets first, then collect like terms. Finally solve for the unknown.
Add or subtract the same inverse both sides of the equation in order to eliminate a number.
Multiply or divide by the same number both sides to get the answer.
The digit on an unknown is called a co-efficient.

Application 11.6

11.7 Solving Fractional Algebraic Equations

Activity

Write a letter of your choice on a slip of paper.

Multiply it by 1/2

Add 4 to the product. The result is 16.
Write an equation relating to the information above.
Solve for the unknown.
What is your answer?
Make a presentation of your working out to the class.

Study tip

When solving equations, apply the concept of operations on integers.
Open brackets, then simplify the equation.
The number or fraction on an unknown is called a co-efficient.
Multiply the co-efficient by all the terms inside the brackets.

Application 11.7

11.8 Solving Problems Involving Equations

Activity

Write an unknown on a slip of paper.
Multiply it by 14.
The result is 64.
Form an equation for the information above.
Try to find the value of the unknown.
Show your working out to the class.

Study tip

Read and understand the word problems in order to form a correct equation.
Apply the correct operations when solving the equation.

Application 11.8

11.9 Solving Algebraic Inequalities with One Unknown

Activity

Solve and discuss the following and show your working steps.
2y + 26 > 6

Study tip

Solve inequalities in the same way you solve equations. But instead of writing the equal sign, write the inequality sign.
When you divide or multiply by a negative number, the < changes to > and > changes to <.

Application 11.9

11.10 Finding the Solution Set

Activity

Write an unknown. Let it be greater than 8.
Multiply it by 2 and add 4.
Solve the inequality formed.
Which numbers are greater than the value of the unknown?
Can you list all of them?
Explain your working out to the class.

Study tip

Solution set is a set of numbers/integers which when substituted in the unknown, make the inequality statement true.
An inequality with one “greater than” or “less than” symbol, gives a set of infinite values to the left or to the right.
Inequalities with two “greater than” or “less than” symbols, give finite setsof values between the two limits.

Application 11.10

Solve and find the solution set:
(a) x > 5 (b) y > 3    (c) 2x + 3 < 11
(d) m < 4 (e) x > 2 (f) 3n - 3 > 6
(g) 4q + 11 < 23    24 > 4y > 8 (i) 6 < 2x < 16
(j) 28 > 4m > 16    (k) 6 < 6x < 6 (l) 96 > 8y > 24

11.11 Solving Problems Involving Simple Algebraic Equations and Inequalities

Activity

Twice a number plus eight gives twenty.
What is the number?
What do you notice?
Discuss and show your working out.

Study tip

To solve problems involving simple equations or inequalities, read the information carefully and interpret it correctly.
Let the unknown number be represented by a letter. Finally, calculate for the value of the letter. Then give the solution set.

Application 11.11

1. I double a number and add six. My answer is less than twelve. Find the number.

2. I multiply three by a number. I get twenty four. What is the number?

3. If one is added to a number the answer is zero. What is the number?

4. If I add 7 to a number, I get ten. What is the number?

5. Think of a number, divide it by two. The answer is four. What is that number?

6. When 6 is added to x, the sum is less than 9. What are the possible values of x?

7. Aidah bought glasses. She broke 5. The remaining glasses are more than 9. What is the possible number of glasses she bought?

8. When a number is multiplied by 5, and reduced by 2, the answer is less than 8. What are the possible values of the number?

End of unit 11 assessment

1. Simplify the following

(a) 10x − 5 + 3x − 4b + 6 (b) –2bcdef − 5bcdef + 12bdecf

2. When I divide a number by five, I get eight. What is the number?

3. If twenty five is subtracted from a number, one is the answer,. What is the number?

4. Find the perimeter of the triangle below:

5. The figure below is a rectangle. Study it and answer the questions that follow.

(a) Find the value of x.

(b) Find the length.

6. James needs a driver who is above 35 years. He should not be above 40 years.

(a) Write an inequality to describe the age limits.

(b) Write the required ages for the job.

7. Solve the following:

(a) -3 < x < 4 (d) 2x + 14 = 0

(b) 2x < 8x + 6 (e) 4x + 7 = 2x + 5

8. A pen costs 3 times as much as a pencil and a book costs 150 Frw more than a pencil. Their total costs is 1,200 Frw. Find the cost of the pencil.

9. Find the number of books that can be shared equally among 6 classes such that each class gets 18 books.

10. Given that; p = 6, q = 4, r = 10, s = 3, evaluate:

(a) 2p + 3s (b) 6r - 4q

11. Given that x = 1/4, y = -6, z = 1/2, evaluate:

(a) 8x - 2y (b) xz

Key unit competence: To be able to use the bearings and compass points and understanding

the relationship between them.
To use the angle sum of a triangle to determine the interior angles of
regular polygons.

Introduction

Polygons are plane shapes with many sides, angles and vertices. Some have
same size of sides and angles while others don’t. The concept of polygon is
used for making different objects. Observe the classroom and identify polygons
shaped objects and try to name them basing on the number of sides and angles.

(a) Is there in your class any object of 3 sides and 3 angles?

(b) Is there any object of 4 sides and 4 angles?

(d) How are the sides of the observed polygons? Find out if they are and name that regular polygon.

12.1 Definition of Polygon and their Examples

Activity

Study tip

A regular polygon has equal sides and equal angles, otherwise it is not regular, it is irregular.
We name a polygon basing on its number of sides.
A circle is not a polygon because it is curved

Application 12.1

12.2 Investigating the Interior and Exterior Angles of a Polygon

Activity

Draw a triangle on a sheet of paper.
Extend the edges from the vertices with straight lines using a ruler.
Measure the inside angle and its adjacent outside angles using a protractor.What names do you give the inside angle and the outside angle?
Find the sum of the two angles you measured?
Now draw a rectangle, pentagon and a hexagon.
Carry out the same steps you carried on a triangle.
What is your observation?
Demonstrate and make a presentation to the class.

Study tip

The inside angle of a polygon is called interior angle.
The outside angle of a polygon is called exterior angle.
The interior angle and exterior angles of a polygon are adjacent to each other.
The interior and exterior angles of a polygon are supplementary angles (add up to 180o)

Application 12.2

Use the polygon cards and sketches. Draw and measure to find the interior and exterior angles of these regular polygons.

(a) Pentagon (b) Septagon (c) Nonagon

(d) Hexagon (e) Duo-decagon (f) Heptagon

(g) Decagon Rectangle (i) Equilateral triangle

(j) Square (k) Nuo-decagon (l) Octagon

12.3 Investigating the Sum of Interior and Exterior Angles of a Regular Polygon

Activity

Get polygon cards.
Trace the edges of a regular pentagon and octagon on sheets of paper.
Extend the edges with a ruler and a pencil to form a straight line.
Using a protractor, measure the interior angles, then add to get their sum.
Then measure the exterior angles, add them to find their sum
What did you find out?
What is the objective of the activity?
Make a presentation to the class.

Study tip

The sum of the interior angles of a regular polygon is found by adding all the interior angles.
All the interior angles of a regular polygon are equal.
The sum of the exterior angles of a regular polygon is found by adding all the exterior angles.
All the exterior angles of a regular polygon are equal.
The sum of all exterior angles of a regular polygon is 360o

Application 12.3

Using the polygon cards, draw and measure to find the sum of the interior
angles and the sum of the exterior angles of the regular polygons below:

(a) pentagon (b) septagon (c) nonagon

(d) hexagon (e) duo-decagon (f) heptagon

(g) decagon rectangle (i) equilateral triangle

(j) square (k) nuo-decagon (l) octagon

12.4 Finding the Interior and Exterior Angles of a Regular Polygon

Activity

Study tip

Interior and exterior angles of a regular polygon are supplementary.
That is, exterior angle + interior angle = 180o.

Application 12.4

1. Find the interior angle given the following exterior angles:

(a) 76o (b) 125o (c) 105o (d) 90o

(e) 140o (f) 65o (g) 109o 123o

2. Find the exterior angle given the following interior angles:

(a) 120o (b) 45o (c) 58o (d) 135o

(e) 65o (f) 125o (g) 85o 144o

12.5 Finding the Sum of Interior Angles of a Regular Polygon

Activity

Draw a regular polygon of four sides.
Measure its interior angles using a protractor.
Record your findings.
Try to measure its exterior angles.
Suggest how you can find the exterior angles while you have interior angles.
Now draw a regular pentagon.
Mark one vertex.
Draw straight lines connecting to other vertices.
What do you observe?

Study tip

Sum of interior angle is equal to each interior angle multiplied by the number of interior angles.
2 is subtracted from the number of sides because 2 sides do not form a triangle.
Sum of interior angle = (n - 2) × 180o where is the number of sides.
Interior angle sum of each triangle is 180o. Number of triangles in a polygonx 180o = interior angle sum of a polygon.

Application 12.5

12.6 Exterior Angles of regular polygons and their Sum

Activity

Study tip

The sum of exterior angles of a regular polygon is 360o.
Number of sides = 360o divided by one exterior angle.

Application 12.6

12.7 Finding Sides and Apothem

Activity

Draw a regular hexagon and mark its centre.
Locate the middle of one side of the polygon.
Draw a straight line from the centre to the point you located on the side
Take a ruler and measure the line.
Explain the procedure you have used and record the result obtained.

Study tip

Application 12.7

1. Differentiate between apothem and side of a regular polygon.

2. Draw a regular nonagon and show the apothem.

3. Draw a regular hexagon and show the apothem.

4. Find the apothem of a pentagon whose side is 8 cm and area is 120 cm2.

5. Find the apothem of a hexagon whose side is 12 cm and area is 144 cm2.

6. Find the apothem of a octagon whose side is 10 cm and area is 160 cm2.

7. What is the apothem of a regular decagon with a side of 8 cm and area of 240 cm2.

12.8 Finding Perimeter of Regular Polygons

Activity

Take a metre ruler.
Measure the length of the sides of the classroom.
Find its perimeter.
State the formula for finding perimeter.
Present your findings to the class.

Study tip

To find perimeter of regular polygons, add the lengths of its sides.

Perimeter is the distance round a shape or polygon.

Application 12.8

12.9 Finding Area of Regular Polygons

Activity

Take a piece of paper and fold it.
Cut it to make 2 equal parts but make sure that the 2 parts are triangles.
Using a ruler, measure their sides.
Find the area of each triangle.

Discuss how you get the area of a regular polygon.

Study tip

Application 12.9

12.10 Finding Bearings and Compass Points

Activity

Study tip

Application 12.10

1. Find the new direction.

(a) Clockwise 135o from West.

(b) Clockwise 180o from North-East.

anti-clowise through 225o. Find her final direction.

(d) Anti-clockwise 90o from East.

12.11 Finding Bearing

Activity

Study tip

To get the bearing, face North and only turn clockwise. Find the total angle of turn.
The angle is written with 3 digits. For example, 10o = 010o.
Degrees of bearing do not involve cardinal points, e.g., 060o not N60oE.

Application 12.11

12.12 Tiling/Construction

Activity

Get sheets of paper.
Use different polygon cards prepared for the lesson.
Using glue, fix suitable polygon cards in a pattern such that no gaps are left in between.
Name the activity you are carrying out.

Which polygon cards can be used to tile the plane? (Rectangular sheet)
How is tiling important in daily life?
Present your working out to class.

Study tip

Tiling is done to cover planes like floors, walls, compounds to form beautiful designs. Square, equilateral triangular, regular pentagonal, hexagonal and other shapes of tiles are used for tiling.

Tiling a plane requires the dimensions of the tiles to fit perfectly the area of the plane.

Designers measure the sides of a plane to calculate the number of tiles to be used.

Application 12.12

1. Use regular pentagonal cards of sides 3 cm to tile a plane sheet of paper.How many cards do you use?
2. Use regular hexagonal cards of sides 2 cm. Tile your table top or desktop.
3. Visit a construction side or buildings. Study the tiling.
(a) What polygons were used?
(b) Copy the tiling design in your books.
4. A room had a floor measuring 9 m by 6 m. A builder laid tiles on the floor measuring 30 cm by 30 cm. How many tiles did the builder lay?
5. A manson laid hexagonal pavers on a compound. Each paver was 40 cm across. The compound was 40 m by 20 m.
(a) Draw a tiling pattern for the above.
(b) How many pavers were laid on the compound?
6. Tiling of square tiles was done on a floor. Each tile had a side of 25 The floor measured 12 m by 10 m.

(a) How many tiles were laid on the floor?
(b) Draw a tiling pattern for the above.

End of Unit 12 Assessment

Key unit competence: To be able to construct polygons using a protractor, a ruler and a pair of

compasses. Design nets to make cuboids and prisms.

Introduction

Boxes, rooms, bars of soap, etc are all cuboids with faces and edges. Some of these cuboids have volume or space to be used as storage of things. For example houses are built with rooms to accommodate people. To build a room of a house,the builders measure the sides of the floor and height of walls suitable to the owner.Can you think on how the area of the faces of one room is measured?

To be concrete, use a box and try to calculate the total surface area of all 6 sides of a box (remember that one face is a rectangular shape). What do you notice?

13.1 Drawing Triangles Using a Protractor and Ruler

Activity

Using a protractor and ruler, draw a right angle, angle 65o and angle 120o.

Explain your working steps.

Present your work to the rest of the class.

example

Step 3: From A, measure 3 cm along the 90o line. Label it D. Do the same at B and label it C.
Step 4: Join D to C and name the square ABCD.

Study tip

A square has 4 equal sides.
Angles of a square measure 90o.
Use a sharp pencil to draw neat and accurate squares.

Application 13.2

Use ruler and protractor to draw the following:
(a) Square PQRS of side 5.5 cm.
(b) Square ABCD of side 4 cm.
(c) Square DEFG of side 3.8 cm.
(d) Square ABCD of side 4.2.
(e) Square WXYZ of side 7 cm.
(f) Square LKMN of side 9 cm.

13.3 Drawing a Rectangle Using a Protractor and Ruler

Activity

Pick a sheet of paper.
Using a protractor, measure the four angles.
Write down the results.
Measure the four sides.
What do you observe?
Make a class presentation.

Study tip

Two opposite sides of a rectangle are equal.
Angles of a rectangle measure 90o each.
Use a sharp pencil and protractor to draw an accurate rectangle.

Application 13.3

Use ruler and protractor to draw the following:
(a) Rectangle ABCD with sides AB = 5.7 cm and BC = 3 cm.
(b) Rectangle DEFG with sides DE = 6.2 cm and EF = 3 cm.
(c) Rectangle HIJK with sides HI = 5.6 cm and IJ = 3 cm.
(d) Rectangle LMNO with sides LM = 6 cm and MN = 4.2 cm.
(e) Rectangle PQRS with sides PQ = 7.2 cm and QR = 4.8 cm.
(f) Rectangle TUVW with sides TU = 7 cm and UV = 4.4 cm.

13.4 Drawing a Regular Pentagon Using a Protractor and Ruler

Activity

Study tip

To construct a pentagon, hexagon and other regular polygons, first find the exterior angle then the interior angle.
Always start with the baseline.
Draw equal sides using ruler.

Application 13.4

Using a protractor and a ruler, draw the following regular pentagons:
(a) Pentagon ABCDE of side 5 cm. (b) Pentagon BCDEF of side 5.5 cm.
(c) Pentagon PQRST of side 6.5 cm. (d) Pentagon ABCDE of side 4 cm.
(e) Pentagon PQRST of side 6.8 cm. (f) Pentagon FGHIJ of side 8.2 cm.

13.5 Drawing a Regular Hexagon

Activity

Pick a pencil, ruler and a sheet of paper.
Without measuring, draw a six sided figure. Name it.
Measure the sides and angles.
What do you observe?
What should you do to draw a regular six sided figure?
Discuss and share with other groups.

Study tip

A regular hexagon has 6 equal sides.
A regular hexagon has 6 interior and 6 exterior equal angles.
Each exterior angle of a regular hexagon is 60^o.
Each interior angle of a regular hexagon is 120^o.

Application 13.5

Using a protractor and ruler, draw the following regular hexagons:

(a) Hexgon ABCDEF of side 4 cm.
(b) Hexgon GHIJKL of side 4.2 cm.
(c) Hexgon KLMNOP of side 5 cm.
(d) Hexgon PQRSTU of side 5.3 cm.
(e) Hexgon DEFGHI of side 3.8 cm.
(f) Hexgon STUVWX of side 4.5 cm.

13.6 Constructing Triangles Using a pair of Compasses and a Ruler

Activity

What is an equilateral triangle?
Using ruler and compasses only, construct an equilateral triangle with side 3 cm.
Present your working out to the class.

Study tip

Use a sharp pencil to construct neat and accurate diagrams.
To construct an equilateral triangle, measure and draw the baseline, then mark the same distances from both ends of the baseline to the top common vertex.
To construct a triangle with all the sides given, measure and draw the baseline accurately. Construct the other two sides using a pair of compasses.
To construct a triangle with 2 sides and 1 angle, construct in the order side,angle, side (SAS).
To construct a triangle with 1 side and 2 angles, construct in the order side, angle, angle (SAA).

Application 13.6

Using compasses and ruler only, construct the following triangles:
1. Triangle ABC of sides 4.5 cm.
2. Triangle PQR with PQ = 6 cm, QR = 4 cm and PR = 3 cm.
3. Triangle XYZ with XY = 5.5 cm, ∠Y = 80o and YZ = 3.5
4. Triangle CDE with CD = 7.5 cm, angle C = 90o and angle CDE = 45o.
5. Triangle JKL with JK = 7 cm, KL = 6 cm and JL = 5 cm.
6. Triangle ABC with AB = 7 cm, AC = 6 cm angle A = 60o.
7. Triangle EFG with ∠GEF = 45o, ∠EFG = 60o and EF = 6.7 cm. Measure EG.

13.7 Constructing a Rectangle Using a pair of Compasses and a Ruler

Activity

Draw line segment XY of 5 cm.
Construct 90o at X and another at Y.
Measure 2 cm from X along the 90o line. Label it point D.
Measure 2 cm from Y along the 90o line. Label it C.
Draw the top side of 5 cm connecting D to C. Name the figure formed.

Study tip

To construct a rectangle, draw the baseline, construct the perpendicular widths and finally draw the opposite length.
Use a sharp pencil and a properly fixed compass.

Application 13.7

Using compasses and ruler only, construct the following rectangles:

(a) Rectangle ABCD with length 5 cm and width 4 cm.

(b) Rectangle WXYZ with length 8 cm and width 5 cm.

(d) Rectangle PQRS with sides 6.5 cm, and 4.2 cm.

(e) Rectangle CDEF with CD = 7.5 cm and DE = 2 cm.

13.8 Constructing a Square Using a pair of Compasses and a Ruler

Activity

Draw a circle on a sheet of paper. Mark the centre.
Cut out the circle from the sheet of paper.
Fold it into 2 equal parts through the centre.
Draw a dotted line along the crease.
Then fold the shape again into 2 equal parts, making quarters.
Open and draw a dotted line along the crease
Draw straight lines connecting the edges where the dotted lines touch the circle.
Cut out the shape formed and measure the sides.
Name the shape you have formed.
Display your findings to the class and make a class presentation. Then poster the cut outs.

Study tip

A square has 4 equal sides.
Use a well sharpened pencil and a properly fixed pair of compasses.
Measure to the accurate cm or mm.
A square has 4 right angles.

Application 13.8

1. Construct squares with radii given below:

(a) 3 cm (b) 4 cm (c) 5.5 cm (d) 7.2 cm

2. Construct a square with a side of 5.6 cm.

3. Construct a square with a side of 5.6 cm.

4. Construct a square with a side of 4.8 cm.

13.9 Finding the Centre Angle of a Regular Polygon

Activity

Study tip

Application 13.9

Find the centre angles of the regular polygons below:
(a) Equilateral triangle (b) Septagon (c) Nonagon
(d) Decagon (e) Duo-decagon (f) Hexagon

13.10 Constructing a Regular Pentagon and Regular Hexagon

Activity

Draw a circle on a sheet of paper.
Mark the centre then draw its radius.
Measure an angle of 72o at the centre.
Draw another radius connecting to the circle. Name the point B.
Place the compass needle at point A and the pencil at B. Using the same length, place at B to make an arc at C.
From C, make an arc at D, then from D, make an arc to make point E.
Using a ruler, connect the points A to B, B to C, C to D, D to E and E to A.
How many sides has the polygon? Name the polygon.
Measure each side. What is the measurement?
Demonstrate and present to the class your procedure and findings.
Carry out the same steps for constructing a regular hexagon but this time use the radius distance to make 6 arcs.
Demonstrate and make a presentation to class.

Study tip

Application 13.10

1. Construct regular pentagons with radii:
(a) 3.5 cm    (b) 4 cm (c) 4.8 cm
2. Construct regular hexagons with radii:
(a) 3.6 cm    (b) 4.2 cm    (c) 4.5 cm
(d) 5 cm (e) 5.6 cm    (f) 8 cm

13.11 Constructing a Regular Septagon and a Regular Octagon

Activity

Draw a circle on a sheet of paper.
Mark the centre then draw its radius.
Using a protractor, draw an angle of 45o at the centre.
Using compasses and pencil, measure the length of the arc along which the two radii from the centre connect.
Make an arc where the 45o line connects the circle.
Do not change the radius of the compass and pencil.
Place the compass point at the previous arc, and make another arc along the circle.
Continue doing the same completely around the circle.
Using a ruler, connect the arcs with straight lines.
How many sides has the figure?
Using a protractor, measure each interior angle. What do you notice>
Write your observations and present to the class.

Study tip

Application 13.11

Using a ruler and compasses, construct the following regular polygons:
(a) Septagon of radius 3 cm.    (b) Septagon of radius 3.6 cm.
(c) Septagon of radius 4 cm.    (d) Octagon of radius 3.8 cm.
(e) Octagon of radius 4 cm.    (f) Octagon of radius 4.2 cm.

13.12 Constructing a Regular Nonagon and Decagon

Activity

On slips of paper, draw a circle of any radius.
Measure a centre angle of 40o.
Using compasses and a pencil, measure the length of arc between the
radius and 40o line. Make an arc.
Without adjusting the compass, draw similar arcs along the circle.
Join the arcs with straight lines.
How many sides has the figure?
What do you observe about its sides?
Measure the interior angles. What do you realise?
Name the figure.

Study tip

Application 13.12

1. Construct regular nonagons with radii:
(a) 3.2 cm    (b) 3.5 cm    (c) 4 cm
(d) 4.3 cm    (e) 5.8 cm    (f) 8 cm
2. Construct regular decagons with radii:
(a) 2.8 cm    (b) 3.4 cm    (c) 4.5 cm
(d) 6 cm (e) 5.6 cm    (f) 10 cm

13.13 Designing Nets of Cuboids, Cubes and Prisms

Activity

Get an empty box of chalk and unfold it carefully.
Display the flat shapes that were folded to form it.
Draw the net of the flat shapes displayed.
Present your working out to the rest of the class.

Study tip

A net is a flat shape that folds up to make a prism.
A prism is a three- dimensional figure with ends that are identical polygons.

Application 13.13

End of Unit 13 Assessment

Key unit competence: To be able to calculate the area enclosed by a circle, surface area of cuboid and volume of a cylinder.

Introduction

Small squares combined form an area of a shape. The area of a circle is made up of small squares of any units of measure, for example, mm, cm etc. Move around noticing surfaces made up of small squares.
Designers, Carpenters, Architects all use the concept of area in their work.
Area depicts the size of space on a flat surface.
(a) Using a ruler, measure the sides of your Math exercise book and find out how many small squares of the units you used has the exercise book surface.
(b) Give examples of where the concept of area is used in real life.

14.1 Finding Area Bounded by a Circle

Activity

Draw a circle of a radius of 3 squares on squared paper.
Estimate the area enclosed by counting the squares.
Add two parts of squares which do not form complete squares to form 1 square.
Practice more with a radius of 2 squares, then 3 squares.
Tabulate the results in the table below:

What do you observe?
Discuss the relationship in the tabulated values.
Share the procedure and outcomes with other groups.

Example 1
Using squared paper, find the area of a circle of radius of 5 squares.

There are 68 whole squares + 20 portions of a square.
Area = 68 squares + 10 squares.
Area = 78 square units

Study tip

1. Using squared paper, find the area of a circle whose radius is:
(a) 3 squares (b) 4 squares (c) 6 squares
(d) 7 squares (e) 8 squares (f) 10 squares
(g) 9 squares 11 squares (i) 12 squares
2. Using the results in number 1 above, complete the table below:

14.2 Calculating Area of a Circle Using Radius

Study tip

14.3 Calculating Area of a Circle Given Diameter

Activity

Pick a circular object.
Measure the diameter.
How can you find the radius?
Calculate area of the circular object.
Share the procedure with your classmate.

Study tip

1. Find the area of a circle whose diameter is 20 cm.
2. A circular garden has a diameter of 40 m. Find its area.
3. Find the area of the centre of a football play ground with a diameter of 4.2 m.
4. A circular billboard has a diameter of 280 cm. Find its area.
5. A tailor made a circular table cloth of a diameter of 1.4 m. Calculate the area.
6. A circular disc has a diameter of 30 cm. Calculate its area?
7. A circular floor has a diameter of 6 metres. What is its area?
8. The diameter of a circular bottom of a tin is 14 cm. Find its area.

14.4 Calculating Area of a Circle Using Circumference

Study tip

1. The circumference of a circle is 88 cm. Calculate its area.
2. The circumference of a circular field is 628 cm. Calculate its area.
3. A circular plot of land has a circumference of 314 m. Calculate its area.
4. A circular stool has a circumference of 78.5 cm. Calculate its area.
5. The circumference of a circular disc is 0.88 cm. What is its area?
6. The circumference of a circular room is 17.6 m. Work out its area.

14.5 Finding Radius Using Area

Study tip

To find radius when area is given, substitute for A and π, then solve.

14.6 Calculating the Surface Area of a Cuboid

Study tip

Surface area is the total area of all the faces of a 3-D shape.
Area of a cuboid = 2lw + 2lh + 2wh
Area of a cube = 6(side × side). A cube has all the sides and faces equal.

14.7 Finding the Length of a cuboid

Activity

Write w = 4, h = 5 and s = 148 on slips of paper.
Try to solve for l in s = 2lw + 2lh + 2wh.
What is the value of l?
Try to formulate other values for w, h and s.

What do you observe?

Study tip

To find length, substitute the width, height and surface area in the formula Surface area = 2(l x w) + 2(l x h) + 2(w x h).

Then solve for length.

14.7 Finding the Length of a cuboid

Activity

Write w = 4, h = 5 and s = 148 on slips of paper.
Try to solve for l in s = 2lw + 2lh + 2wh.
What is the value of l?
Try to formulate other values for w, h and s.
What do you observe?

Study tip

To find length, substitute the width, height and surface area in the formula Surface area = 2(l x w) + 2(l x h) + 2(w x h).
Then solve for length.

14.8 Finding the Width of a cuboid

Study tip

To find width, substitute the length, height and surface area in the formula Surface are = 2(l x w) + 2(l x h) + 2(w x h).
Then solve for width.

14.9 Finding Height of a Cuboid
Activity

On slips of paper, write a = 6, b = 7 and y = 292.
Try to solve the equation: y = 2ab + 2ac + 2bc.
What is the value of c?
Show your working out to the class.

Study tip

To find height, substitute the width, length and surface area in the formula Surface area = 2(l x w) + 2(l x h) + 2(w x h).
Then solve for height.

14.10 Finding Volume of a Cylinder

Study tip

Key unit competence: To be able to extend methods for collecting data, and representing and interpreting it in order to answer the questions or explore hypothesis.

Introduction
Data are helpful in knowing how things are taking place. They are also helpful in good planning. With records, one is able to remember what took place in the past.
People are able to know things that happened long ago due to stored data or information. Business people, schools, government record data or information.
(a) How do you think collecting data and keeping them is important in daily life?
(b) Which kind of data can a person collect? Give an example of data which can be collected in the environment.
(c) Is correcting data, organizing data and interpreting them useful to make a family budget? What do you think about a national budget?
(d) Do you think statistics is important to you? Give 2 reasons.

15.1 Collecting Data to Investigate a Question

Activity

List the marks obtained by every learner in your class in the end of month test.
Group the same marks, and record their number.
Use strokes to represent the number of marks recorded.
Put these in a table form.
Explain the data in the table to the class.

Example
Below is mass in kg of P. 6 learners who underwent a Body Mass Test.
28 30 28 33 35 40 40 28 30 30 42 40 35 40 40 40 33 30
28 30 30 30 33 40 33 33 35 40 35 40 35 35
(a) Use tally marks to organise the data above in the table
(b) How many learners underwent the Body Mass Test?

Study tip

Frequency is the number of occurrences.
Collecting data using tables helps in grouping data.
Investigating data helps in analysis and proper planning.

1. The following marks were obtained by a P. 6 class in a Mathematics Examination.
55 68 66 66 83 57 70 73 70 73 55 66 68 83 66
73 72 57 73 83 57 71 65 78 74 56 68 71 83 85
83 68 70 71 55 70 61 63 67 72 70 74 59 60
(a) Use tallies to organise the data in a table.
(b) How many learners did the examination?
2. Kayitesi bought kg of maize flour for several days as shown below.
5 3 6 5 6 3 1 2 1 2 3 4 5 6 3 4 4 3 5 2 1 1 2 3 4 5 6 4 5 6
6 6 5 4 4 2 3 4 3 4 2 2 4 1 4 2 3 4 4 3 2 4 6 5 4 6 4 5 4 5
(a) Organise the data in a table.
(b) State the number of kg she bought altogether.

15.2 Interpreting Data in Frequency Tables

Example

The table below shows the marks obtained by P.6 learners in a Mathematics exam by percentage in a school. Study it and answer the questions that follow.

(a) How many learners did the exam? 60 learners

(b) Which mark was scored more by learners? 81%

(d) How many learners scored 70%? 3 learners

(e) Find the number of learners who got above 80%. 46 learners

Study tip

Frequency is the number of times a quantity appears.
Tallies are written with strokes to show how many times a number appears.
Tallies are also used to show the occurrences.

1. Mr. Kagame planted different types of saplings. The data is shown in the

table below. Study it and answer the questions that follow.

(a) How many cypress and podo saplings were planted?

(b) Which type of saplings had the same number?

(d) How many saplings were planted altogether?

2. The table below shows the age of learners in P. 6 class. Use it to

answer the questions that follow:

(a) How many learners are 12 years old?

(b) How old are most of the learners?

15.3 Representing Data in a Bar Chart

Activity

The table below shows items that a country exports as well as the contribution of each item to the total item value.

Represent the above information in a bar graph.

Make a presentation to the class.

Example

The table below shows number of learners per class in a school in Rwanda.

(a) Represent the data in a bar chart.

(b) How many learners are in the whole school?

Solution

(a) A bar graph showing number of learners per class in a school

(b) 45 + 40 + 50 + 60 + 55 + 45 = 295. The school has 295 learners.

Study tip
## A bar graph is a graph consisting of vertical or horizontal bars whose lengths are proportional to the amounts or quantities of statistical data represented.
## A bar graph is also called bar chart or bar diagram.
## Always choose a suitable scale for the vertical axis.

1. The table below shows the number of litres of milk collected on a farm in a week.

(a) Represent the above information in a bar graph.

(b) How many litres of milk were produced in the week?

2. P. 6 learners were asked to pick clothes of their best colour and the results were recorded as follows: G = green, R = red, B = blue, O = orange and Y = yellow.

(a) Organise the data in a table.

(b) Represent the data in a bar chart.

15.4 Interpreting Data in a Bar Chart

Activity

 Study the bar graph below and answer the questions that follow:

How many days are shown on the graph?

What is the scale on the vertical axis?

Find the number of trays collected in the whole week.

On which day did Mr. Uwera collect fewer trays of eggs?

Example

1. The bar graph shows Mr. Muhire’s maize exports.

(a) In which months did Mr. Muhire export the same number of tonnes?
(b) How many more kg of maize were exported in November than in August?
(c) Find the total tonnes of maize exported in the last half of the year.
Solution
(a) In the months of September and November
(b) November = 5 tonnes and August = 2 tonnes
(5 − 2) tonnes = 3 tonnes.
but 1 tonne = 1,000 kg
3 tonnes = 3 × 1,000 kg = 3,000 kg
Therefore 3,000 kg more were exported in the month of November
than in August.
(c) Total tonnes exported = (4 + 2 + 5 + 1 + 5 + 3) tonnes = 20 tonnes.

Study tip
## The title tells the subject on which data was collected. Always take note
of what is represented on the vertical and horizontal axes.
## The vertical axis is divided into equal parts.

## The reading on each part of the vertical axis is the vertical scale.

Study the graph and answer the questions that follow:

1. Beans in kg sold at a shop in 5 weeks

(a) In which weeks did the shopkeeper sell the same quantities of beans?
(b) In which week did the shopkeeper sell the highest quantity of beans?
(c) Find the total kilogram of beans sold in the 5 weeks
(d) If each kg of beans costs 700 Frw, how much did the shopkeeper
get from the sale of beans in the five weeks?

2. The bar graph below shows the number of textbooks bought by a primary
school in 2016. Study it and answer the questions that follow:

(a) Which subject has the highest number of copies?
(b) Which subjects have the same quantity?
(c) How many more English books were bought than Social Studies books?
(d) If each Mathematics book costs 5,500 Frw, how much money was
spent on the Mathematics textbooks?
(e) How many textbooks were bought?

15.5 Representing Data in Pie Charts

Activity

P. 6 learners at a school performed as follows in the end of year examination; Division I, 18 learners, Division II, 25 learners, Division III, 12 learners and Division IV, 5 learners.

Get the fraction of each division.

Multiply each fraction by 360^o to get the degrees of each division.

Draw a circle and divide it into sectors by using the corresponding degrees.

What special name is given to the above circle?

Present your working out to the class.

Study tip

15.6 Interpreting Data in Pie Charts to Draw a Conclusion

Study tip
## To get the number of items for each sector, multiply the fraction for that
sector by the total number.
## To get percentage for each sector, multiply the fraction for the sector by 100%.

End of unit 15 assessment

1. Below are heights in cm of P.6 learners in their study of measurement.
125 128 118 130 142 125 125 118 118 128
134 128 118 125 130 128 130 142 125 128
130 128 130 132 128 132 130 125 142 128
130 128 130 128 140 134 130 128 132 138
(a) Use tally marks to organise the data above in a table.
(b) How many learners were measured?
(c) What was the modal height?

2. The following are age groups of people in a village on the Voter’s register.

Represent the data in a bar chart.
3. P.6 learners were asked to state the food type they like most. 9 learners
like cassava, 10 learners like rice, 5 learners like Irish potato, 6 learners
like banana. Represent the information in a pie chart.
4. In 2016, some people were asked to predict the national team which would win the World Cup. Their predictions were as follow:

Represent the above information in a pie graph and in a bar graph.

5. In a village, there are different games played. Some fans were asked which of these game they like most and the responses were recorded as follows: 60 like football, 22 like netball, 19 like tennis, 44 like basketball and 35 like volleyball. Represent the information in a pie chart?

(a) What is the scale on the vertical axis?
(b) On which day did Domina collect 56 litres?
(c) Find the total of litres collected on Thursday and Friday?
(d) On which days did Domina collect the same litres of milk?
(e) If each litre costs 800 Frw, how much did Domina receive when she sold the milk she collected on Wednesday?

Key unit competence: To be able to order events in terms of likehood (impossible, equally likely, certain).

Introduction
Things in nature have no law of happening. Some things occur by chance. Some things must happen; some have even chance of happening, while others are practically impossible. For example, before any Football match starts, the spectators are not sure who will win. The chance of winning the game is equal at the begging.
At the end of the game, different possible outcomes or results may occur.
(a) With support of examples could you predict how the different possible
   outcomes or results will look like?
(b) Do you think that to have the same score in a football match is certain
   to happen? Equally likely to happen? or impossible to happen? Justify
   your answer.

16.1 Vocabulary of Chance: Impossible, Certain, Equally Likely, Events,

Activity

Play a game of Bingo or any other game.
Write down six numbers between 1 and 12 of your choice.
Toss two dice. Strike out the number shown by the dice if it is in your chosen number list.
If your chosen number list has all the strikes, call out Bingo.
So, are some numbers easier to get than others? Why?
Is the appearance of a number a chance? Discuss.

Study tip

Probability is defined as the chance that something will happen.
Probability is written as numbers between zero and one.
If the probability is 1 then the event is certain. That is, it is going to happen.

For example,when you toss a coin, a head or tail shows up.

The probability of 0 means that an events is impossible. If you toss a coin you cannot get both a tail and a head at the same time, so this has 0 probability.

Likely means that the probability of one event is higher than the probability of another event.
Unlikely means that one event is less likely to happen.
Equal probability means that the chance of each event to happen are equal.

1. Discuss the likelihood of the following events:
(a) Getting a head when you toss a coin.
(b) It will rain in Kigali this year.
(c) A woman will give birth to a boy.
(d) The sun will rise tomorrow.
(e) You will get a six when you throw a die.
(f) You will get a total of 1 when you throw two dice.
(g) Our teacher will become the president.
I was born yesterday.

2. Explain the following word used in probability:
(a) Impossible
(b) Certain
(c) Equally
(d) Likely
(e) Unlikely
3. Create your own statements and associate them with vocabulary of chance.

16.2 Using Data to Decide How Likely Something is to Happen

Activity

Roll a die (remember there are 6 options, that is; (1, 2, 3, 4, 5, 6)
How many ways are there of a 2 showing up?
How many ways are there of a 7 showing up?
Make a presentation to class.

Study tip
## Probability is always greater than or equal to 0 and less than or equal to 1.
## Probability means how possible something may happen.
## Probability is how likely it is that something will happen.

4. Two coins are tossed once. Find the probability that head only is obtained.
5. A card is drawn at random from a deck of cards. Find the probability of
getting the king of a heart.

End of Unit 16 Assessment
1. What is the probability of the following events:
(a) A coin tossed showing 6.
(b) Raining today.
(c) A mother giving birth to a baby boy or girl.
(d) A cow eating grass.

2. The table below shows the learners enrolment in a school.

Use the language of chance to define these statements.
(a) The school started in 2010.
(b) Enrolment in 2018 will increase.
(c) The learners in the school perform well.
3. Suppose you roll a die. Find the probability that the number obtained is
less than 4.
4. A card is drawn at random from a deck of cards (52 cards). Find the probability
of getting the king of diamond.
5. There are red, blue and black pens in a pack. What is the chance of
picking at random:
(a) a black pen?
(b) a blue pen?
(c) black, blue or red pen?
(d) Order the likelihood of chances to happen.

ASSESSMENT File

Mathematics

General

Unit1:Reading, Writing and Comparing Whole Numbers Beyond 1,000,000

Unit1:Reading, Writing and Comparing Whole Numbers Beyond 1,000,000

1.3 Finding Place Value and Values of Numbers up to 7 Digits

Activity

1.4 Comparing Numbers Using <, > or =

1.5 Arranging Numbers in Ascending and Descending Order

Key unit competence:

Introduction

9.1 Calculating Simple Interest

Study tip

Unit 9: Simple Interest and Problems Involving Saving

Activity

If rate is a fractional percentage, change it into a common fraction first

Application 9.2

9.3 Solving Problems Involving Simple Interest

Activity

Study tip

Application 9.3

9.4 Calculating Interest Rate

Activity

Study tip

Application 9.4

9.5 Solving Problems Involving Interest Rate

Activity

Study tip

Application 9.5

9.6 Calculating Principle

Activity

Study tip

Application 9.6

9.7 Solving Problems Involving Principle

Activity

Study tip

Application 9.7

9.8 Calculating Time

Activity

Study tip

Application 9.8

9.9 Solving Problems Involving Time

Activity

Study tip

Application 9.9

9.10 Calculating Amount

Activity

Study tip

Application 9.10

9.11 Solving Problems Involving Amount

Activity

Study tip

Application 9.11

9.12 Saving Money in the Bank or Putting it in Investments

Activity

Study tip

Application 9.12

9.13 Solving Problems Involving Savings

Activity

Application 9.13

End of unit 9 assessment

Key unit competence: To be able to write sequences of whole numbers, fractions and decimals.

10.1 Algebraic Expressions

Application 10.1

10.2 Equivalent Expressions

Application 10.2

10.3 Finding the Missing Consecutive Numbers

Activity

10.4 Finding the Missing Consecutive Fractions and Decimals

Activity

Application 10.4

10.5 Finding the General Term/rule of a Linear Sequence

Application 10.5

10.6 Finding the General Term/rule of Linear Sequence for Fractions and Decimals

Application 10.6

10.7 Finding the Missing Number or nth Term in a Linear Sequence

10.8 Finding the Missing Fraction or nth Term in a Linear Sequence

Application 10.8

10.9 Finding the Number Sequence Using the General Term/rule

Application 10.9

End of Unit 10 Assessment