Direct proportion

Direct ProDirect Propportion.ortion.

100o

O

A B

5cmL

T

o

D 0 3 6 10 20

E 24

Direct ProDirect Propportion All Around.ortion All Around.Direct proportion applies to all of the situations below:

In each of these examples two quantities are mentioned. Can you identify them in each example.

Number of Mars bars. Cost of bars.

Hours you work. Pay

Dollars. Pounds.

Area of sector. Angle of sector.

The number of hours you work and the pay you earn.

The number of dollars you get for your pounds on holiday.

The number of mars bars you buy and the price you pay. M

ars

The area of a sector of a circle and the angle at the centre .

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The FindinThe Findingg One Method. One Method.Example 1.

If it costs 85p for 5 Mars bars, what is the cost of 3 Mars bars ?

Solution.

Find the cost of 1 Mars bar.Cost of 1 mars bar :

85 5 = 17 p

Find the cost of 3 Mars bars.Cost of 3 Mars bars.

17 x 3 = 51p

3 Mars bars cost 51p.

Example 2.

If you are paid £261 for 18 hours work, how much will you be paid for 23 hours work ?

Solution.

Find the pay for 1 hour .I hours pay

261 18 = £ 14.50

Calculate 23 hours pay.14.5 x 23 = 333.5

You are paid £333.50 for 23 hours work.

Example 3.

It takes a production line 90 hours to make 120 cars. How long does it take to make 330 cars ?

Solution.

Calculate the time to make 1 car .Time for 1 car.

90 120 = 0.75 hours. Note :0.75 hours = 45 minutes .

Calculate the time to make 330 cars.0.75 x 330 = 247.5 hours

It takes 247 hours and 30 minutes to make 330 cars.

What Goes In The Box ? What Goes In The Box ? Calculate the following:

(1) It costs £21.06 for a hotel to buy 27 boxes of cereal . How much will 35 boxes cost at the same price per box ?

(2) 35 cars are produced in 87.5 hors by a car factory. How long will it take to produce 142 cars at the same rate of working ?

(3) If you earn £139.20 for 16 hours of work, what will you earn for 9 hours work paid at the same rate ?

£27.30

355 hours

£78.30

The Cross-MultiThe Cross-Multipplication Method .lication Method .This method is a more sophisticated way of solving direct proportion questions but it has two advantages:

• It establishes a very strong routine to solve the problems.

• It makes Inverse Proportion questions easier to handle.

Think back to the Mars Bar problem and answer the following questions:

If you have twice as many Mars Bars what happens to the price ?

The price doubles.

If you have three times as many Mars Bars what happens to the price?

The price trebles.

What about ten times the number of Mars Bars? Ten times the price.

Key Point.

Whenever you have two quantities and when one is doubled or trebled etc the other quantity doubles or trebles etc , then you have direct proportion .

The problem can then be solved by the method demonstrated in the following slides.

Example 1.

If you are paid £ 190.21 for 23 hours work, then what is your pay for 17 hours work ?

Solution. Identify the two quantities being talked about:Pay HoursIdentify the two amounts that go together.190.21 23

Ask yourself “ If the first quantity is doubled , trebled etc, what happens to the second quantity ?

It doubles , trebles etc also.

Determine what is being asked for and what other info’ you have.

P 17

Cross multiply to solve.

Use arrows..

17

23

p

190.21

23 x p = 17 x 190.21

23

190.2117p

p = £140.59 for 17 hours.

Example 2.

If 5 hectares of wheat take 12 hours to harvest, how long will 27 hectares take ?

Identify the two quantities being talked about:

Hectares Time

Identify the two amounts that go together.

5 12

Determine what is being asked for and what other info’ you have.

27 T

Ask yourself “ If the first quantity is doubled , trebled etc, what happens to the second quantity ?

It doubles , trebles etc also.

Use arrows..

Cross multiply to solve.

5

27=

12

T

5 X T = 12 x 27

T =

12 x 27

5

T = 64.8 hours = 6hr 48min

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Example 3.Find the area of the sector OAB of this circle if angle AOB is 100o

Solution.

100o

O

A B

5cm

Find the area of the whole circle.

A = r2

A = 3.14 x 5 x 5

A = 78.5cm2

Now decide on the two quantities in the problem:

Area Angle

What angle goes with 78.5cm2 ?

78.5 360o

Add the other information:

A 100o

Cross multiply and solve:

100

360

A

78.5

A = 21.8cm2 ( to 1 d.p)

If you double treble etc the angle what happens to the area ?

Example 3.

172o

O

A

B12cm

Calculate the arc length AB on this circle.

Solution.

Find the circumference of the circle.

C = 2 r

C = 2 x 3.14 x 12

C = 75.36 cm

Now decide on the two quantities in the problem:

Arc Length Angle

What angle goes with 75.36cm ?

75.36 360o

A 172o

If you double treble etc the angle what happens to the arc length ?

Cross multiply and solve:

172

360

A

75.36

A = 36cm (to 1 d.p)

What Goes In The Box ? 2What Goes In The Box ? 2(1) The shadow of a tree 7.4m high is 10.6m . How long will the shadow of a tree 8.9m high be at the same time ?

(2) Calculate the area of the sector AOB . 100o

O

A B

23cm

(3) Calculate the arc length AB.

(4) You get 14 out of 23 in a test. What would you get out of 100 for the same test score ?

12.7m

461.4cm2 to 1 d.p

40.1cm to 1 d.p

60.9% to 1 d.p

GraGrapphs Of Direct Prohs Of Direct Propportion.ortion.Consider the following Direct Proportion situation :

It takes 2 hours for a company to manufacture each ladder that it makes:

Complete the table below:

Ladders 0 1 2 3 4 5 Hours 0 2 4 6 8 10

Now draw a graph of the above table:

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0 2 4 6 8 10 12

2

4

6

8

10

No of ladders.

Hours

Graph Of Ladders Made against Time Taken.

L

T

o

Key Fact.

All Direct Proportion graphs are straight line graphs passing through the origin:

Look at the table of values again:

Ladders 0 1 2 3 4 5 Hours 0 2 4 6 8 10

If you divide the number of hours by the given number of ladders , what do you find each time ? ( Hours divided by ladders)

The answer is 2 every time.

We only need one point on the table to complete the whole table.

Complete the tables below given that they form direct proportion graphs:

(1) F 0 1 2 3 4 5 G 80 4 12 16 20

(2) D 0 3 6 10 20

E 240 18 40 80

(3) R 0 2 7 10 100

W 350 10 50 500

(4) W 0 3 7 8 12

M 120 4.5 10.5 18

0 1 2 3 4 5

10

20

30

G

H

What Goes In The Box ? 3What Goes In The Box ? 3

Complete the tables below for direct proportion and draw the graph.

G 0 1 2 3 4 5

H 180 6 12 24 30