Mathematics

Key unit competence

By the end of this unit, I will be able to use a multiplier for proportional change.

Unit outline

• Proportions and sharing.

• Expressing ratios in their simplest form.

• Sharing quantities in a given proportion.

• Increasing and decreasing quantities by a given percentage.

• Calculations on proportional change using multiplier.

Introduction

In S1,we already learned about proportion, its definition, properties and its application in the real life situation. This unit, therefore, reviews the properties of proportions; expressing ratios in the simplest form;  and decreasing quantities by a given percentage proportion. Finally, calculations of proportional change using multiplier.

4.1 Properties of proportions

In S1, we already studied proportions, its properties and application in real life situation. Here, we review some of the properties of proportions.

Activity 4.1

1. Working in groups, discuss what proportion is.

2. By use of a simple example, discuss within your group members the properties of proportions that you learnt in S1.

3. Compare your findings with the other groups in class. Did you get the same findings?

4. Discuss two applications of proportions in real life.

5. Present your findings to the whole class through your group leader.

The following are properties of proportions: 1. Mean-extremes or cross-multiplication property. If

 a /b =  c /d,

then ad = bc 

4.2 Expressing ratios in their    simplest form

Remember that expressing ratios in their simplest form had been studied in S1.  You are therefore expected to be well versed with how the operations will be carried out.

Activity 4.2

• Remind you classmate on how ratios are expressed in their simplest form.

• Consider the ratio 4:12. What do you notice when the ratio is divided by 4 on both sides?

• Explain your findings to your partner.

As studied in S1, simplification of ratios is a means of reducing a ratio to its lowest form by either dividing or multiplying the ratio by the same value without necessarily changing the value of the ratio.

Exercise 4.1 1.

Express the following ratios in their lowest forms.

(a)  28 : 42 

(b)  30 : 50

(c) 24 kg : 30 kg

(d) 150 cm to 3 m

(e)  1 litre to 250 ml

(f)  45 min : 1 1 2 hours

(g)  2.6 kg to 130 g 

  160 cm3 to 2 litres

2.  Simplify the following ratios in their simplest form.

(a)  2 : 0.4                                              (b)  0.9 : 0.18
(c)  0.3 : 0.12 (d) 3/4 : 10
(e)  3/4 : 3/5   (f) 3(1/2) : 2(1/2)

4.3 Multipliers for proportional change

4.3.1 Definition of multiplier

Activity 4.3

• Discuss with your classmate what you understand by the word multiplier.

• Consider a shirt that is sold at a 20% discount.

• What is the percentage of the selling price?

• Convert this percentage you have gotten into fraction. What do you notice?

• Consider the price of a book being reduced by 15%, the percentage of the selling price is  100% – 15% = 85%. • 85% converted to decimal gives

  = 85/100 = 0.85 • We say that 0.85 is the multiplier of the price of the book.

Example 4.8

What is the multiplier for 15% increase?

Solution

A 15% increase means the final percentage for the quantity will be 100% + 15% = 115% 115% as a decimal = 115/100 = 1.15    1.15 is the multiplier.

Example 4.9

What is the multiplier of 45% decrease?

Solution

45% decrease means the overall percentage for the quantity will be 100% – 45% = 55% 55% as a fraction =   55/100 = 0.55    0.55 is the multiplier.

4.3.2 Multiplier for increasing and decreasing by a percentage

Activity 4.4

• Consider a phone costing 10 000 FRW. Two customers bought the phone at two different places. One at 20% less while the other at 20% more than the cost price.

• Determine the prices at which the customers bought the phones

. • Compare your results with those of other groups

4.3.2.1 Increasing multiplier

Exercise 4.2

Increase:

(a) 50 by 10%

(b) 60 by 30%

(c) 70 by 5% 

(d) 200 by 80%

(e) 450 by 100% 

(f) 525 by 25%

4.3.2.2 Decreasing multiplier

Exercise 4.3

Decrease

(a) 200 by 20%

(b) 150 by 5%

(c) 450 by 35%

(d) 670 by 45%

(e) 1 000 by 3%

(f) 1 425 by 25%

4.4 Calculations of proportional  change using multiplier

Activity 4.5

Consider a shirt with a marked price of 500 FRW. After bargaining with the customer, the shirt is sold at a 10% lower. Discuss with your classmate the change in price and the new price (selling price) of the shirt in FRW.

From Activity 4.5, it is evident that the shirt has been sold at a reduced price compared to the initial buying price. The marked price is reduced proportional by 10% which translates to 50FRW. Therefore the customer bought the shirt at 50 FRW less.

Exercise 4.4 1.

Increase:

(a) 70 by 20%

(b) 250 by 50%

(c) 750 by 100%

(d) 1 250 by 5%

(e) 2 by 95%

(f) 100 by 0.75%

2. Decrease:

(a) 600 by 20%

(b) 30 by 30% (

c) 1 760 by 10%

(d) 230 by 11%

(e) 980 by 99%

(f) 2 250 by 2%

3. Mbaya bought 10 m of cloth material for making suits. After washing the length shrunk by 5%. What was the length after washing?

4. The bus fare from Town A to Town B used to be 600 FRW. Due to increase in petrol, the fare has increased by 25%. What was the new fare?

5. Habimana’s salary used to be  45 000 FRW. The company started making losses and his salary was reduced by 15%. What is his new salary?

Unit summary

1. A ratio is a relation that compares two or more quantities of the same kind, such as lengths, using division giving one quantity as a fraction of another.

2. A proportion is a mathematical statement of the equality of two ratios.

3. The four properties of proportion are: (a) Mean-extremes or cross - product (b) Mean or extremes switching (c) Inverse or reciprocal (d) Denominator addition/     substraction

4. If two ratio have the same value then they are equivalent, even though they may look different.

5. A decreasing multiplier is a factor that reduces the proportion of a given quantity. To calculate the new price, we proceed as  New price = initial price × multiplier,   where,  multiplier = (100 - x) /100 and x is the    percentage decrease on the cost price.

6. An increasing multiplier is a factor that increases the proportion of a given quantity. To calculate the new price, we proceed as  New price = initial price × multiplier,   where  multiplier = (100 + x)/100 and x is the    percentage increase on the cost price.

Unit 4 test

1. Express the following ratios to their simplest form.

(a) 8:24 

(b) 0.2:0.8

(c) 1 8 : 1 2 

(d) 3 4 : 20

2. Uwimana has a flock of 3 000 sheep. He intends to reduce the flock by 40%. What number will be his new flock?

3. A cow produced 800 litres of milk in one week. In the following week its milk production increased by 30%. What amount of milk did it produce in the week.

4. The attendance in an agricultural shows that last year was 60 000 people. This year the attendance increased by 12%. What was the attendance this year?

5. A farmer produces 9 500 tonnes of coffee in the first six months of the year. Because of drought in the following six months, the production reduced by 36 %. How many tonnes of coffee were produced in the year?

6. In 2005, a certain region increased milk production by 22% over the previous year. If in 2004 the region had produced 25 450 000 litres of milk, how many litres were produced in the two years?

7. The population of a town increases by 8% every year. The population this year is 52 800. What will be the population after 2 years?

8. In 2003, the number of HIV and AIDS patients visiting a certain dispensary fell by 25%. If a total of 1200 patients had visited dispensary in 2002, how many patients visited the dispensary in 2003?

9. A milk processing company has a capacity to process 6 million litres of milk in 2 months. In the last two months, processing fell by 45% due to repair undertaken in the factory. How many litres of milk were processed in the two months?

10. A region produces 5 million tonnes of maize every year. The production fell by 15% in the following years. How many tonnes were produced by the region over the two years?

11. Three business partners Patric, Rebecca and Joseph contributed  50 000 FRW, 40 000 FRW and  25 000 FRW respectively, to start a business. After sometime, they realised a profit which they decided to share in the ratio of their contributions. If Joseph’s share was 10 000 FRW, by how much was Patric’s share more than Rebecca’s?