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)
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
