Solve proportion problems (direct, inverse, proportional to the square, inversely proportional to the square)
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 x^{3}, y = kx^{3}
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(2^{3}) = k8
Rearrange to get: k = 1/8
So, y = 1/8 x^{3} for all values of x and y.
So if y = 8, we multiply by 8 to get x3
64 = x^{3}
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 x^{2}.
If y = 2, and x = 4, what is x when y = 1?
Using the formula for indirect proportion: 2 = k/4^{2}
Rearrange to get k: k = 32
So, y = 32/x^{2} for all values of y and x.
So if y = 1: 1 = 32/x^{2}
Rearrange to find x: x^{2} = 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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