Direct Direct ProportionProportion

Direct Proportion Graphs

Direct Proportion formula and calculations

Inverse Direct Proportion

Direct Proportion

Other Direct Proportion formula

Direct Direct ProportionProportion

“ .. When you double the number of cakes you double the cost.”

Cakes Cost

Two quantities, (for example, number of cakes and totalcost) are said to be in DIRECT Proportion, if :

Direct Proportion

Example : The cost of 6 cakes is £4.20. find the cost of 5 cakes.

6 4.20 1 4.20 ÷ 6 = 0.70 5 0.70 x 5 = £3.50

Write down two quantities that

are in direct proportion.

Direct Direct ProportionProportion

£ $

Direct Proportion

Example : On holiday I exchanged £30 for $45.How many $ will I get for £50.

30 45 1 45 ÷ 30 = 1.5 50 1.5 x 50 = $75

What name do we give to

this value

Exchange rate

Direct Direct ProportionProportion

(a) Simply multiple by 2 :

Direct Proportion

Example : To make scrambled eggs for 2 people we need 2eggs, 4g butter and 40 ml of milk.How much of each for(a) 4 people (b) Just himself

4 eggs, 8g butter, 80 ml of milk.

(b) Simply half original amounts:

1 eggs, 2g butter, 20 ml of milk.

Direct Direct ProportionProportion

The table below shows the cost of packets of “Biscuits”.

Direct Proportion Graphs

No. of Pkts 1 2 3 4 5 6

Cost (p) 20 40 60 80 100 120

We can construct a graph to represent this data.

What type of graph do we expect ?

Apr 19, 2023Apr 19, 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.

Direct Proportion

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6

No. of Packets

Cos

t (p

)

Direct Proportion GraphsNotice that the points lie on a straight line

passing through the origin

This is true for any two quantities

which are in Direct Proportion.

Direct Direct ProportionProportion

Direct Proportion Graphs

KeyPoint

Two quantities which are in Direct Proportion

always lie on a straight linepassing through the origin.

Direct Direct ProportionProportion

Example : Plot the points in the table below. Are they in Direct Proportion?

Direct Proportion Graphs

We plot the points (1,3) , (2,6) , (3,9) , (4,12)

XX 11 22 33 44

yy 33 66 99 1212

1

Direct Direct ProportionProportion

Plotting the points

(1,3) , (2,6) , (3,9) , (4,12)

Direct Proportion Graphs

0 1 2 3 4

101112

23456789

Since we have a straight linepassing through the origin

x and y are in Direct Proportion.

x

y

1

Direct Direct ProportionProportion

Find the formula connectingy and x.

Direct Proportion Graphs

0 1 2 3 4

101112

23456789

Formula has the form :

x

y

Gradient = 3

Formula is :y = 3x

y = kx

Direct Direct ProportionProportion

Direct Proportion Graphs

Important facts:Fill in table

Find gradient from graph

Write down formula using knowledge from straight line chapter

Direct Direct ProportionProportion

Q. Given that y is directly proportional to x, and when y = 20, x = 4.

Find a formula connecting y and x

Direct Proportion Formula

Since y is directly proportional to x the formula is of the form

y = kx k is the gradien

t20 = k(4)

k = 20 ÷ 4 = 5y = 5x

Direct Direct ProportionProportion

Q. The number of dollars (d) varies directly as the number of £’s (P). You get 3 dollars for £2.

Find a formula connecting d and P.

Direct Proportion Formula

Since d is directly proportional to P the formula is of the form

d = kP k is the gradien

t3 = k(2)

k = 3 ÷ 2 = 1.5d = 1.5P

Direct Direct ProportionProportion

Q. How much will I get for £20

Direct Proportion Formula

d = 1.5P

d = 1.5 x 20 = 30 dollars

Direct Direct ProportionProportion

Q. Given that y is directly proportional to the square of x, and when y = 40, x = 2.

Find a formula connecting y and x when .

Since y is directly proportional to x squaredthe formula is of the form

y = kx2

40 = k(2)2

k = 40 ÷ 4 = 10y = 10x2

Harder Direct Proportion Formula

y

x

Direct Direct ProportionProportion

Q. Calculate y when x = 5

y = 10x2

y = 10(5)2 = 10 x 25 = 250

Harder Direct Proportion Formula

y

x

Direct Direct ProportionProportion

Q. The cost (C) of producing a football magazine varies as the square root of the number of pages (P). Given 36 pages cost 45p to produce. Find a formula connecting C and P.

Since C is directly proportional to “square root of” P the formula is of the form

k = 48 ÷ 6 = 8

Harder Direct Proportion Formula

C =k P48 =k 36

C =8 P

y

x

Direct Direct ProportionProportion

Q. How much will 100 pages cost.

Harder Direct Proportion Formula

C =8 100

C =8 100 =8×10 =80p

y

x

Inverse Inverse ProportionProportion

Inverse Proportion is when one quantity increasesand the other decreases. The two quantities are said

to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other.

Example : Fill in the following table given x and yare inversely proportional.

Inverse Proportion

XX 11 22 44 88

yy 8080 102040

y

x

Inverse Inverse ProportionProportion

Men Hours

Inverse Proportion is the when one quantity increasesand the other decreases. The two quantities are said

to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other.

Example : If it takes 3 men 8 hours to build a wall.How long will it take 4 men. (Less time !!)

3 8 1 3 x 8 = 24 hours

4 24 ÷ 4 = 6 hours

Inverse Proportion

y

x

Inverse Inverse ProportionProportion

Men Months

Example : It takes 10 men 12 months to build a house.How long should it take 15 men.

10 12 1 12 x 10 = 120

15 120 ÷ 15 = 8 months

Inverse Proportion

y

x

Inverse Inverse ProportionProportion

Speed Time

Example : At 8 m/s a journey takes 32 minutes.How long should it take at 10 m/s.

8 32 mins 1 32 x 8 = 256 mins

10 256 ÷ 10 = 25.6 mins

Inverse Proportion

y

x