### Direct and Inverse Proportion Questions

How do you solve GCSE Maths questions about direct and indirect proportion?

Here's a little how to of the different techniques needed to help you master proportion problems.

### Direct Proportion

If two quantities are directly proportional, then as one increases, the other increases by the same percentage.

Suppose the two quantities are y and x, we can show that they are directly proportional by writing

### y = kx

where k is some constant which we call the constant of proportionality.

### Example of direct proportion problem.

Suppose y and x are directly proportional.

If y = 3, and x = 12, what is x when y = 5?

First, you need to find k.

Using the formula for direct proportion: 3 = k(12)         (as outlined above)

Rearrange to get: k = 3/12 = 1/4

So, y = 1/4 x for all values of x and y.

So if y is 5, we divide by 1/4 to find x.

Dividing by 1/4 is the same as multiplying by 4.

So if y = 5, x = 20

### Example with direct proportion to powers.

If x is to the power, we can use the same method.

Suppose y is directly proportional to x3, y = kx3

If y = 1, x = 2, what is x when y = 64?

First, you need to find k.

Using the formula for direct proportion: 1 = k(23) = k8

Rearrange to get: k = 1/8

So, y = 1/8 x3 for all values of x and y.

So if y = 8, we multiply by 8 to get x3

64 = x3

x = 4

### Inverse Proportion

If two quantities are inversely proportional, then as one increases, the other decreases by the same percentage.

Suppose the two quantities are y and x, we can show that that are inversely proportional by writing

### y = k/x

where k is some constant which we call the constant of proportionality.

### Example of inverse proportion problem

Suppose y is inversely proportional to x (NOTE: x inversely proportional to y is the same thing)

If y = 2, and x = 4, what is y when x is 8?

First, you need to find k.

Using the formula for indirect proportion: 2 = k/4

Rearrange to get k: k = 2x4 = 8

So, y = 8/x for all values of y and x.

So if x is 8, we divide 8 by 8 to get y.

So if x = 8, y = 1

x has increased (by double) while y has decreased (by half)!

### Example with indirect proportion to powers.

Suppose y is inversely proportional to x2

If y = 2, and x = 4, what is x when y = 1?

Using the formula for indirect proportion: 2 = k/42

Rearrange to get k: k = 32

So, y = 32/x2 for all values of y and x.

So if y = 1: 1 = 32/x2

Rearrange to find x: x2 = 32

x is the square root of 32.

Hopefully these questions and answers about direct and indirect proportion will help you understand, and if you need a little practice then just have a go at the 'Test Questions' where you'll find some past papers with proportion problems.

You could also ask a question to the community and see if someone can help you out with a specific problem you're struggling on.

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