How to work out direct and inverse proportion

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Here are some notes on how to work out direct and inverse proportion.

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Direct proportion

T is proportional to S, so we write T=kS
k is the constant of proportionality.

If T is directly proportional to S

When T=24 and S=6
Find T when S=18
and find S when T=3

Substitute the values into our equation
24=k6 this becomes 24= k x 6

So if we rearrange the equation and do 24/6 our value for k is 4.

Now we can substitute again so, 4 x 18=? ?= 72 T=72
and, 3= 4 x ? so we do, 3/4= 0.75  S= 0.75

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Direct proportion

Using powers

M is directly proportional to L^2
When M =1 L= 2
Find M when L=4

Our equation is M= kL^2

So we need to sub in our values;

1= k x 2^2        1= k x 4      1/4=0.25 so k=0.25

Sub in the values again;

M= 0.25 x 4^2  M= 0.25 x 16  M=4

Then, because L is to the power of 2 we have to square root the value of M. The square root of 4 is 2 so when L=4, M=2

Sorry it might be a bit hard to follow but it is quite easy to understand once you get the hang of it.

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Direct proportion

If you get a question where one letter is proportional to the cube of a letter

e.g. O is directly proportional to Y^3

You basically do the same working out as on the last page but instead of square rooting the end result you cube root it.

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Inverse proportion

The constant of proportionality is k

Y is inversely proportional to X, so we write Y=k/X but if we rearrange it becomes XY=k

When Y=3 X=12
Find the value of X when Y=8

Sub our values in;

3 x 12= 36 so k=36

Sub in again;

X x 8=36 do some rearranging to find the value of X. X=4.5

So when Y=8, X=4.5

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Inverse proportion

Again, inverse proportion can involve you to work out powers

A is inversely proportional to B^3

When B=2 A=25
Find the value of B when A=60


Sub in the values

25 x 2^3 =  200

Sub in our values to work out the answer to our question;

200 = 60 x B^3
we rearrange B^3= 200/60 so B^3= 3.333

Now we need to cube root that value so it becomes 1.494

So B=1.494 when A = 60 

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Very helpful especially the powers bits 


A good range of proportion questions with solutions

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