If you’re looking for the answers to your ‘direct proportion’ questions, then you’ve come to the right blog! The concept of direct proportion is a relatively simple one – as one value increases, the other increases at the same rate. Even though this seems relatively simple, it can be applied to a lot of different situations and the types of questions you can get on a GCSE paper will vary a lot.

Let’s consider two variables, 𝑥 and 𝑦, that are directly proportional. If 𝑥 is doubled, then 𝑦 is also doubled. If 𝑥 is halved, then 𝑦 is halved. In fact, if we multiply or divide one of the variables by any number, the other variable will be changed in the same way.

We can plot the two variables on a set of axes. Whenever the two variables are directly proportional, the graph will be a straight line that passes through the point (0, 0). We can use this relationship to find the value of one variable when we know the value of the other. One way to do this is to first find the value of 𝑥 when 𝑦 = 1 or of 𝑦 when 𝑥 = 1. This is called the unitary method.                             

Since 2 exercise books cost £2.50, we can divide this by 2 to find the cost of 1 exercise book.

2 exercise books = £2.50

1 exercise book = 2.50 ÷ 2
                        = £1.25

Now we can multiply the cost of 1 exercise book by 5 to find the cost of 5 exercise books.

5 exercise books = 1.25 × 5
                          = £6.25

Now we can subtract this from the total cost to find the cost of 3 pens and repeat the process to find the cost of 10 pens.

3 pens = 7.36 – 6.25
         = £1.11

1 pen = 1.11 ÷ 3
         = £0.37

10 pens = 0.37 × 10
         = £3.70

a. Since 24 is a multiple of 12, we don’t need to find the amount of flour to make one cookie. Instead, we can find the amount of flour needed by doubling the amount of flour needed for 12.

250 × 2 = 500g

If you prefer, you could still find the amount of flour for 1 cookie and multiply it by 24 – it’s just not the most efficient method.

b. For this part, it’s easier to find the number of cookies that we can make using 1 egg.

3 eggs = 12 cookies
1 egg = 12 ÷ 3
         = 4 cookies
10 eggs = 4 × 10
         = 40 cookies

Now that you have answers to your direct proportion questions, you’re probably ready for more revision. You can find more of our blogs here! You can also subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too.