Cambridge International Mathematics

Two variables aredirectly proportionalif multiplying one of them by a number

results in the other one being multiplied by the same number.

In theOpening Problem, the variablesCandnaredirectly proportional.

We say thatCis directly proportionalton, and write C/n.

The variables are connected by the formula or law C=2n since 2 is the gradient of the line.

If two quantitiesxandyaredirectly proportional, we write y/x.

The symbol/reads is directly proportional to or varies directly as.

If y/x then y=kx wherekis a constant called theproportionality constant.

Whenyis graphed againstxthenkis thegradientof the graph, and the line passes through theorigin.

Example 1 Self Tutor

Fruit buns cost 60 cents each. Supposexbuns are bought and the total price is$y.

a Show, by graphical means, thatyis directly proportional tox.

b Find: i the proportionality constant ii the law connectingxandy.

a x^012345

y 0 0 : 60 1 : 20 1 : 80 2 : 40 3 : 00

Since the graph is a straight line passing through

(0,0), y/x.

biGradient=

0 : 60 ¡ 0

1 ¡ 0

=0: 6

So, the proportionality constantkis 0 : 6.

ii The law connectingxandyis y=0: 6 x.

Example 2 Self Tutor

Suppose y/n andy=40whenn=3. Findnwheny= 137.

Method 1:

Since y/n, y=kn wherekis the

proportionality constant.

Since n=3when y=40, 40 =k£ 3

)^403 =k

) y=^403 n

So, when y= 137, 137 =^403 n

) 137 £ 403 =n

) n¼ 10 : 3

Method 2:

n 3?

y 40 137

To changeyfrom 40 to 137 , we multiply by^13740.

Since y/n, we also multiplynby^13740

) n=3£^13740 ¼ 10 : 3

A DIRECT VARIATION [2.13]

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606 Variation and power modelling (Chapter 30)

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