Mathematics SSE

Key Unit competence: Use arithmetic operations to solve simple real life problems

1.0. Introductory Activity 

The simple interest earned on an investment is I prt = where I is the interest earned, P is the principal, r is the interest rate and t is the time in years. Assume that 50,000Frw is invested at annual interest rate of 8% and that the interest is added to the principal at the end of each year.

 a. Discuss the amount of interest that will be earned each year for 5 years. 

b. How can you find the total amount of money earned at the end of these 5 years? Classify and explain all Mathematics operations that can be used to find that money.

 1.1 Operations of real numbers and their properties 

1.1.1. Sets of numbers

a. Carry out research on sets of numbers to determine the meanings of natural numbers, integers, rational numbers and irrational numbers. Use your findings to select the natural numbers, integers, rational numbers and irrational numbers in the list of the numbers given below:

b. From (a) deduce the definition of the set  of real numbers. 

c. Given any 3 real numbers a b and c , of your choice, discuss the following:

Application activity 1.1.1

1.1.2 Properties of operations in the set of real numbers

Activity 1.1.2

Application activity 1.1.2

 1) Use the distributive property to simplify

1.2 Fractions and related problems

Activity 1.2

An algebraic fraction exists only if the denominator is not equal to zero. The values of

the variable that make the denominator zero is called a restriction on the variable(s).

An algebraic fraction can have more than one restriction.

Examples: 

Problems related to fractions

Example 

Application activity 1.2

1.3 Decimals and related problems 

Activity 1.3

CONTENT SUMMARY

Decimals are just another way of expressing fractions

1.4 Percentages and related problems

Activity 1.4

Application activity 1.4

1. In the middle of the first term, the school organize the test and the tutor of

mathematics prepared 20 questions for both section A and B. peter get 80%

correct. How many questions did peter missed?

2. Student earned a grade of 80% on mathematics test that has 20 questions. How

many did the student answered correctly? And what percentage of that not

answered correctly?

3. John took a mathematics test and got 35 correct answers and 10 incorrect

answers. What was the percentage of correct answers?

4. As a future teacher, is it necessary for a student teacher to know how to

determine the percentage? Explain it with supportive examples. 

1.5 Negative numbers and related problems

Activity 1.5

1. The temperature of a juice in the bottle was 0 20 C . They put this juice in the

fridge so that its temperature decreases by 0 30 C . What is the temperature of

this juice? What can you advice the child who wishes to drink that juice? 

2. Suppose that you have a long ruler fixed from the hole and graduated such

that the point 0 corresponds to the ground level as illustrated on the following

figure.

What is the coordinate of the point position for an insect which is at 3 units below

the ground level in a hole? 

CONTENT SUMMARY

There are numerous instances where one comes across negative quantities, such

as temperatures below zero or bank overdrafts. For example, if you have 3500Frw

in your bank account and withdraw 6000Frw with an acceptable credit, your bank

balance is -2500Frw . There are instances, however, where it is not usually possible to

have negative quantities. For example, a firm’s production level cannot be negative.

From the above activity, you have learnt that, you can need to use negative or a

positive numbers.

For example, when measuring temperature, the value of the temperatures of the body

or surrounding can be negative or positive. The normal body temperature is about

0 +37 C and the temperature of the freezing mercury is about 0 −39 C 

Example:

a) Eight students each have an overdraft (scholarship advance) of 21,000Frw. What

is their total bank balance?

Solution:

The total balance in the bank is 8 × (−21,000) = −168,000Frw. The sign negative

means that students have the credit to be paid. 

Application activity 1.5

Question1: Answer to the following questions

c) Where negative numbers are applied in the real life? Do you think that

computers of bank managers deal with negative numbers when operating loans

for clients?

Question2:

A cylinder is ¼ full of water. After 60ml of water is added the cylinder is 2/3 full.

Calculate the total volume of the cylinder.

Question3:

Between 1990 and 1997 the population of an Island fell by 4%. The population in

1997 was 201,600. Find the population in 1990.

1.6. Absolute value

1.6.1 Meaning of absolute value

Activity 1.6.1

1) Draw a number line and state the number of units found between

CONTENT SUMMARY

Activity 1.6.2

Evaluate and compare the following:

Application activity 1.6

b) Dr. Makoma went from Kabgayi to Muhanga City on foot at the constant speed

of 100m/min. If he used 60 minutes to go and 60 minutes to come back. Explain

to your colleague the distance and the displacement covered by Dr. Makoma. 

1.7. Powers and related problems

Activity 1.7

1. How can you find the area of the following paper in the form of a square?

2. Given a cube of the following form: 

Determine the volume of this cube. 

CONTENT SUMMARY

1.7.1 Meaning of power of a number

Thus, Compound interest = Accumulated amount (A) – Principal (P)

Note that the principal and the interest earned increased after each interest period.

We can also deduce that I =A-P 

Examples 

Application activity 1.7

1) Simplify

1.8 Roots (radicals) and related problems

1.8.1 Meaning of radicals

Activity 1.8.1

1) Evaluate the following powers using a calculator

Example 

Example: 

Application activity 1.8.2

1.8.3 Operations on radicals

Activity 1.8.3

Simplify the following

Addition and subtraction

When adding or subtracting the radicals we may need to simplify if we have similar

radicals. Similar radicals are the radicals with the same indices and same bases.

Example:

1.8.4 Rationalizing radicals

Activity 1.8.4

Make the denominator of each of the following rational

Example: 

Application activity 1.8

1.9 Decimal logarithms and related problems

Activity 1.9

1) What is the real number at which 10 must be raised to obtain:

a) 1 b) 10 c) 100 d) 1000 e) 10000 f) 100000

Application activity 1.9

1. Without using calculator, compare the numbers a and b.

1.10 Important applications of arithmetic

Activity 1.10

Make a research in the library or on internet and categorize problems of Economics

and Finance that are easily solved with the use of arithmetic.

Focus on the following: Elasticity of demand, Arc of elasticity for demand, Simple

interest and compound interest, Final value of investment. 

CONTENT SUMMARY

1. Elasticity’ of demand and Arc of elasticity for demand

Elasticity’ of demand:

Price elasticity of demand is a measure of the responsiveness of demand to changes

in price. It is

usually defined as

Although a positive price change usually corresponds to a negative quantity change,

and vice versa, it is easier to treat the changes in both price and quantity as positive

quantities. This allows the (−1) to be dropped from the formula. The 0.5 and the 100

will always cancel top and bottom in arc elasticity calculations. 

Example:

Calculate the arc elasticity of demand between points A and B on the demand

schedule shown

In the following figure: 

Solution:

Between points A and B price falls by 5 from 20 to 15 and quantity rises by 20 from

40 to 60. Using the formula defined above: 

2. Simple interest

Simple interest is the amount charged when one borrows money or loan from a

financial institution which accrue yearly.

This interest is a fixed percentage charged on money/loan that is not yet paid.

This interest is calculated based on the original principal or loan and is paid at regular

intervals

4. Calculating the final value of an investment

 Consider an investment at compound interest where:

P is the initial sum invested, A is the final value of the investment, r is the interest

rate per time period (as a decimal fraction) and n is the number of time periods.

The value of the investment at the end of each year will be 1+ r times the sum

invested at the start of the year.

Thus, for any investment,

5. Time periods, initial amounts and interest rates

The formula for the final sum of an investment contains the four variables F, A, i and 

n.

So far we have only calculated F for given values of A, i and n. However, if the values

of any three of the variables in this equation are given then one can usually calculate

the fourth.

Initial amount

A formula to calculate A, when values for F, i and n are given, can be derived as

follows.

Since the final sum formula is

Time period

Calculating the time period is rather more tricky than the calculation of the initial amount.

Application activity 1.8.2

1. An initial investment of £50,000 increases to £56,711.25 after 2 years. What

interest rate has been applied?

1.11. END UNIT ASSESSMENT

1. Why is it necessary for a student teacher to study arithmetic? Explain your

answer on one page and be ready to defend your arguments in a classroom

discussion;

2. The price of a house was 2000000Frw in the year 2000. At the end of each year

the price has increased by 6%.

a) Find the price of the house after one year

b) Find the price of that house after 3 years

c) Find the price that such a house should have in this year.