Direct proportion

If two quantities are in direct proportion, as one increases, the other increases by the same percentage.

If y is directly proportional to x, this can be written as y ∝x

A simple example of two things that are in the same proportion is the amount of apples you might buy and the amount you pay for them. If you buy twice as many apples as your friend, you pay twice as much.

We can write the connection between the cost and the amount as an equation:

Cost of apples = price per apple × number of apples bought.

This can also be written as y = kx, where k is the cost (the price per apple).

This means that, for some constant k, y = kx for all values of x and k is called theconstant of proportionality.


If y is directly proportional to x.

When x = 12 then y = 3

Find the constant of proportionality and the value of x when y = 8.

We know that y is proportional to x so y = kx

We also know that when x = 12 then y = 3

To find the value of k substitute the values y = 3 and x = 12 into y = kx

3 = k × 12

So k = 3/12 = 1/4

To find the value of x , when y = 8 substitute y = 8 and k = 1/4 into y = kx

8 = (1/4) x

So x = 32 when y = 8

Direct proportion to powers

y can be directly proportional to x2 , x3 and other powers of x.

They can always form an equation with k, a constant multiplier (the constant of proportionality), at the start.

eg y = kx2


y ∝ x 3

If y = 1 when x = 2, find the value of y when x = 4


y ∝ x 3

So y = kx3

Substitute the value y = 1 and x = 2 into y = kx3 to find the value of k.

1 = k × 23

So k = 1/8

Now use the values k = 1/8 and x = 4. y = 1/8 × 64

Gives the answer y = 8

Inverse Proportion

Inverse proportion is when one value increases at the other value decreases.

A simple example of inversely proportional quantities is the lengths and widths of rectangles with the same area. As the length of one side doubles, the width has to be halved for the area to stay the same.

rectangles measuring 3cm by 2cm and 6cm by 1cm


y is inversely proportional to x. When y = 3, x = 12 .

Find the constant of proportionality, and the value of x when y = 8.

y ∝ 1/x

y = k/x

So xy = k

Substitute the values x = 12 and y = 3 into xy = k

3 × 12 = 36

So k = 36

To find the value of x when y = 8, substitute k = 36 and y = 8 into xy = k

8x = 36

So x = 4.5

Again, you can have questions involving squares, cubes or other powers of the variables.


v is inversely proportional to r3. When r = 2, v = 25. Find r when v = 60.


v = ∝  so v =

Re-arrange the above to get k on its own.

k = vr3

k = 25 × 23

So k = 200

When v = 60

60r3 = 200

r3 = 200/60

r3 = 3.333

So r equals the cube root of 3.333

So r = 1.494

Graphical Representation

When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In simpler terms, that means if A is always twice as much as B, then they directly vary. If a gallon of milk costs $2, and I buy 1 gallon, the total cost is $2. If I buy 10 gallons, the price is $20. In this example the total cost of milk and the number of gallons purchased are subject to direct variation — the ratio of the cost to the number of gallons is always 2.

To be more “mathematical” about it, if y varies directly as x, then the graph of all points that describe this relationship is a line going through the origin (0, 0) whose slope is called the constant of the variation. That’s because each of the variables is a constant multiple of the other, like in the graph shown below:

Graph of a straight line illustrating direct variation

Inverse Variation
(The Opposite of Direct Variation)

In an inverse variation, the values of the two variables change in an opposite manner – as one value increases, the other decreases.

For instance, a biker traveling at 8 mph can cover 8 miles in 1 hour.  If the biker’s speed decreases to 4 mph, it will take the biker 2 hours (an increase of one hour), to cover the same distance.

Inverse variation:  when one variable increases,
the other variable decreases.

Notice the shape of the graph of inverse variation.
If the value of x is increased, then y decreases.
If x decreases, the y value increases.  We say that y varies inversely as the value of x.

An inverse variation between 2 variables, y and x, is a relationship that is expressed as:

Y = k/x

where the variable k is called the constant of proportionality.

As with the direct variation problems, the k value needs to be found using the first set of data.

The Reciprocal of a Number

Clearly, 3 X 1/3 = 1

1/3 is called the reciprocal of 3

3 is called the reciprocal of 1/3

One number is the reciprocal of another if their product is 1.

Example, the reciprocal of 5/6 is 6/5 since 5/6 X 6/5 = 1.

In general:

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator, i.e. by inverting the fraction.


Find the reciprocal of 20.


Reciprocal of 20 is 1/20

Find the reciprocal of 3/7.


Example 9


To find the reciprocal of a mixed number, change it into an improper fraction and then invert it.