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

If **y is directly
proportional to x**, then the formula connecting y and x is **y = kx**.

Direct proportion means as x increases then so does y.

**k is known as the
constant of proportionality** and this is found by substituting the boundary
conditions into the formula.

Let’s take a look at an example:

**Example 1**

If y is directly proportional to x, and when x =3 y = 18, find a formula for y.

Since this example is direct proportion, then your formula is:

y = kx

All you need to do now is substitute the boundary conditions (x=3 and y = 8) and solve the equation to find k:

18 = k × 3

k = 6 (divide both sides by 3)

Since you have now worked out k, you can substitute k back into the original formula to give:

y = 6x

The graph of y = 6x is shown below.

**Example 2**

If c is directly proportional to the square of d, and when d =8, c = 16, find a formula fo c.

Again this is direct proportion so the formula is:

c = kd² (make sure you put d² and not just d)

All you need to do now is substitute the boundary conditions (d=8 and c = 16) and solve the equation to find k:

16 = k × 8²

16 = k ×64

k = 16/64 = 0.25 (divide both sides by 64))

Since you have now worked out k, you can substitute k back into the original formula to give:

c = 0.25d²

Below is the graph of c = 025d²

**Example 3**

If a is directly proportional to the square root of b, and when b = 100, a = 40, find a formula for a. Also calculate the value of a when b = 9.

Again this is direct proportion so the formula is:

a = k√b (make sure you put √b and not just b)

All you need to do now is substitute the boundary conditions (b=100 and a = 40) and solve the equation to find k:

40 = k × √100

40 = k ×10 (divide both sides by 10)

k = 4

Since you have now worked out k, you can substitute k back into the original formula to give:

a = 4√b

Now since you have found the formula for a, you can now use the formula to work out the value of a when b = 9. All you need to do is sub b = 9 into the formula you have just found.

a = 4√b

a = 4 × √9

a = 4 × 3

a = 12

For some help on inverse proportion check out this hub:

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

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