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LINEAR EQUATIONS

Linear equations are always seen to be abstract, but in this lesson you will discover how to remove that.

RATES, RATIO, PERCENTAGES AND PROPORTION

PROPORTION: DIRECT AND INDIRECT

BACKGROUND INFORMATION

Rates:

This is an expression of the relationship between two quantities for example

1. Population growth rate expresses the relationship between

the population and the elements of time.

2. Speed is the relationship between distance and time, S=D/T

Ratio:

This involves comparing quantities where units are the same.

Ratios are normally given in their simplest form, for example

In a class 18 boys and 24 girls we can say that the ratio of

boys to girls in the class are 18:24 . This ratio can be reduce to 3: 4 by dividing both numbers by 6

Any ratio is expressed in the form new value: old value

**EXERCISE 1. **

Which of the following are directly proportional?

b) <![endif]> The amount of sugar and the total cost

c) <![endif]> The height of the students and their mathematics grades

3.

journey of 240 km cost assuming the fare is charged in

proportion to distance ?

number of students is reduced to 48 how many toilets papers

are likely to be used in a term?

LESSON OBJECTIVE(S)

By the end of the lesson you should be able to;

recognise direct and indirect proportions from given situations.

solve problems involving direct and indirect proportions

correctly.

QUIZ

1. A car travels 72 km on 6 six litres of petrol .how much petrol will it consume to travel a distance of 600 km?

2. A train travels a at a speed of 50 km/hr for 9 hours. At what speed should it move to cover the same distance in 7 hours and 30 minutes

3. 20 men working 8 hours a day recarpeted a section of a road in 20 days how many hours a day must 50 men work inorder to recarpet similar section of the road in 10 days?

4. A group of 40 girls finished a piece of work in 20 days .

How many extra girls are needed to complete the same

work in 16 days?.

5. Susan used sh 120 to buy 24 oranges .How many oranges

would she have bought with sh 300.

RATES, RATIO, PERCENTAGES AND PROPORTION

Introduction

Proportion is a relationship between two quantities such that changes

in one quantity cause a change in the other quantity.

There are two types of proportions;

. Direct proportion

. Indirect (inverse) proportion

Direct Proportion

Direct proportion is where two quantities increase or decrease at the same rate,

for example if one mango costs sh. 20, two mangoes will cost ksh.258 the

number of mangoes increases as the total cost increases.

Thus the number of mangoes is directly proportional to their total

cost. Ksh 20, Ksh 40

Example 1

Jane drives 60km in 40 minutes .How far will she in 2 hours

and 30 minutes at the same speed?

Solution

The distance traveled is directly proportional to the total distance taken.

The time increases from 40 minutes to 150 minutes.

So the ratio of increase in time is 150: 40 = 15: 4

The distance also increases in the same ratio: i.e.

New distance = 15/4 * old distance

=15/4*60 = 15*15 = 225 km

Example 2

In a secondary school, 8 teachers are needed to teach 120 students.

(a) How many teachers will be needed to teach 300 students?

(b) How many students will require 12 teachers?

Solutions.

(a) In this case, the number of teachers is directly proportional

(b) to the number of students.

The number of students has increased from 120 to 300.

In the ratio, 300: 120 = 5: 2

So the number of teachers = 5/2 * old number of teachers

=5/2 *8 = 5*4 = 20 teachers.

Question (b)

How many students will require 12 teachers?

Solution

(b). The number of teachers increase 8 to 12 in the ratio 12 : 8 = 3 : 2

Hence new numbers of students increase in the same ratio so new

number of students i.e. 3/2 * 120 = 3*60 = 180 students.

Example 3.

20 oranges cost shs. 60. How much will 10 oranges cost?

Solution.

The number of oranges decreases from 20 to 10 in the

ratio 10 : 20 = 1 : 2 So the total cost decreases in the same ratio.

New cost = 1/2 * old cost.

= 1/2 * 60 = Shs. 30

**EXERCISE 2 **

1.A race cyclist takes 54 minutes for a race if he cycles at 48 km/hr.

A what speed must he cycle to do the race in 36 minutes?

2. Three tractors plough a farm in 6 six days. How long will take to

plough the farm using two tractors

3. A piece of land has enough grass to feed 20 cows for 5 days how

long can feed a) 1 cow b) 4 cows c) x cows

4. Given that x and y are inversely proportional. Copy and complete

the table below

How long will the food last if 50 more refugees are brought in?

INDIRECT/INVERSE PROPORTION

Indirect proportion is where one quantity increases while the other

decreases at the same rate and vice versa

For example in a certain farm 5 people can dig a plot of land in 10

days. if the number of people is increased to 10 (doubled),then the

number of days will decrease to 5 days (halved).

The number of people is indirectly(inversely) proportional to the total

time taken to finish the work.

Which of the following pairs of quantities are inversely proportional?

a) A mans height and his age.

b)Number of people needed to dig a farm and the time taken to finish it

c)The time taken to travel a

fixed distance and speed.

d Distance travelled by a car and

amount of petrol consumed.

e) Number of units of an

article bought for a fixed amount

of money,cost per unit of article

SOLUTIONS

Inverse proportions relations are

b,c, and d

Example 2

Jane drives from Nairobi to Voi for 5 hours at

90km/hr, how long

does she take if she drives at 75km/hr?

Speed is reduced as time increases. This is inverse proportion.

speed decreases in the ratio 75:90=5:therefore time increases in

the ratio 6:5 hence new time=6/5*old time

=6/5*5=6 hours

A family of 6 has enough food to last 12 days how long will the food

last if the family receives 3 visitors?

Number of people increases from 6 to 9 in the ratio 3:2. the number

of days will decrease in the ratio 2:3. Hence the food will last

2/3 *12 = 8 days

If quantities x and y are inversely proportional, when x increases

in the ratio a:b y decreases in the ratio b:a

RATES AND PROPORTIONS

Our daily life involves rates and proportions of differents things

Learn how how useful they are

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