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)

More by this Author


Comments

No comments yet.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.


    Click to Rate This Article
    working