© Daniel Holloway

If two quantities are in direct proportion,
as one value increases, the other will
increase by the same percentage

With inverse proportion, one value
increases as the other decreases
An example of direct proportion is buying
produce. If you were to buy one sandwich
at 45p, you would have to pay 90p for two.
We can show this direct proportionality with
an equation:
cost of
sandwiches
=
selling price
per sandwich
x
number
purchases
This can also be written as y = kx where k is
the price per apple. Whenever k is used in
proportion – this means the constant value
When y is directly proportional to x we can
write y ∝ x
 Worked Example
If y is directly proportional to x when x = 12, y = 3
Find the value of x when y = 8
We know that y is proportional to x:
y = kx
Substitute the values into the formula
3 = k x 12
k = 3/12 = ¼
To find the value of x when y = 8:
8 = ¼x
x = 8 x 4 = 32
y can be directly proportional to the powers
of x (i.e. x2, x3), which also forms an equation:
y = kx2
 y ∝ x3
y is directly proportional to the cube of x
if y = 1 when x = 2
what is the value of y when
x = 4?
An example of inversely proportional
quantities is the changing lengths and
widths of rectangles with the same area. As
the length of the rectangle doubles, the
width has to be halved in order for the area
to remain the same.
3cm
6cm
2cm
6cm2
1cm
6cm2
 Worked Example
y is inversely proportional to x. When y = 3, x = 12
a) Express k in terms of x and y
b) Find the value of x when y = 8
a Because y is inversely proportional to x, we say:
y ∝ 1/x
which we change to
y = k/x
Therefore yx = k
b Substitute the values of y = 3 and x = 12 in the expression:
3 x 12 = 36 so k = 36
To find the value of x when y = 8,
substitute y = 8 and k = 36 into yx = k
8x = 36
x = 4.5
Test Yourself
1) y is inversely proportional to the square of x
and when y = 9, x = 2
a Find y in terms of x
b Find (+)x when y = 144
2) The resistance, R to the motion of a sailing
boat is directly proportional to the square of
its speed, s
If R = 300 when s = 12, find the value of s
when R = 150
3) The wavelength, w metres, is inversely proportional to the
frequency, f kHz, of the waves
a A radio wavelength of 1000 metres has a frequency of 300kHz
The frequency is doubled to 600kHz
What is the new wavelength?
b Calculate the frequency when the wavelength is 842 metres
c Radio 1 has a frequency value which is equal to its
wavelength value
Calculate the frequency and wavelength of Radio 1 in kHz
and metres
4) y is inversely proportional to the square root of x
When y = 6, x = 4
a What is the value of y when x = 9?
b What is the value of x when y = 10?