Cambridge International Mathematics

If two painters can paint a house in 3 days, how long would it take four painters working at the same rate

to paint the house?

The four painters will be able to do twice as much work in the same amount of time. So, it will take them

a half of the time, or 112 days.

The two variables in this example are thenumber of paintersand thetime taken.

In a case like this where doubling one variable halves the other, we have aninverse variation.

Two variables areinversely proportionalorvary inverselyif, when one is

multipliedby a constant, the other isdividedby the same constant.

Dividing bykis the same as multiplying by

1

k

, so we can see that if one of the variables is multiplied by

2 , the other must be multiplied by^12 ,orishalved.

Consider again the example of two painters completing a job in

three days.

Supposexis the number of painters andyis the number of days

to complete the job. Clearlyxmust be an integer.

This table shows some of the possible combinations. They are

plotted on the graph alongside.

x 1 2 3 4 6

y 6 3 2 112 1

The shape of the graph is part of ahyperbola.

Now consider the graph ofyagainst

1

x

.

x 1 2 3 4 6

1

x

(^112131416)
y 6 3 2 112 1
Notice that the points on this graph form a straight line which
passes through the origin.
So, ifyisinversely proportionaltox, thenyisdirectly proportionalto

1

x

.

Consequently, y=k£

1

x

or y=

k

x

or xy=k.

In the painters example, notice that xy=6 for all points on the graph. So, in this case k=6.

B INVERSE VARIATION [2.13]

5

5

x

y

O

1

5

y

(^1) –
x
O
612 Variation and power modelling (Chapter 30)
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Y:\HAESE\IGCSE01\IG01_30\612IGCSE01_30.CDR Monday, 27 October 2008 2:57:44 PM PETER