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# Direct and inverse proportion examples with graphs (inversely and directly proportional formulas)

In this hub I will explain the difference between inverse proportion and direct proportion.

If **m is directly
proportional to n** then the formula is:

**m = kn **

If **m is inversely
proportional** to n then the formula is:

**m = k/n**

k is known as the constant of proportionality. Notice with direct proportion you multiply k by n and direct proportion k is being divided by n.

Let’s take a look at an example:

a) If y is directly proportional to x, and when x = 6 y = 3, find a formula connecting y and x.

b) This time y is inversely proportional to x, find a formula for y using the same boundary conditions as part a).

**Answers**

a) Since y is directly proportional to x then the formula is going to be y = kx.

The next thing you need to do is sub in the values x = 6 and y = 3 into this formula to calculate the constant of proportionality:

3 = k × 6

So to find k, all you need to do is divide the left hand side of the equation by 6:

3/6 = k

0.5 = k

Since k has now been found, you can plug this back into the formula that you started off with:

y = 0.5x

b) This time y is inversely proportional to x then the formula is going to be y = k/x.

Like part a), the next thing you need to do is sub in the values x = 6 and y = 3 into this formula to calculate the constant of proportionality:

3 = k ÷ 6

So to find k, all you need to do is multiply the left hand side of the equation by 6:

3 × 6 = k

18 = k

Since k has now been found, you can plug this back into the formula that you started off with:

y = 18/x

If you plot these two formulas on a graph you will notice that part a is an increasing formula and part b is a decreasing formula as x increases.

For some more help on direct and inverse proportion check out these hubs:

Direct proportion. How to write down a formula if y is directly proportional to x. Maths GCSE.

Inverse proportion word problems with graphs ( the meaning of y is inversely proportional to x)

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