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 β 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.

Letβs consider two variables, π₯ and π¦, that are directly proportional. If π₯ is doubled, then π¦ is also doubled. If π₯ is halved, then π¦ 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 π₯ when π¦ = 1 or of π¦ when π₯ = 1. This is called the **unitary method**.

**Example 1**

3 pens and 5 exercise books cost Β£7.36.

2 exercise books cost Β£2.50.

How much will 10 pens cost?

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

2 exercise books = Β£2.50

1 exercise book = 2.50 Γ· 2

= Β£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 Γ 5

= Β£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 β 6.25

= Β£1.11

1 pen = 1.11 Γ· 3

= Β£0.37

10 pens = 0.37 Γ 10

**= Β£3.70**

**Example 2**

Here is a list of ingredients for making 12 cookies.

**a.**How much flour will Pedra need to make 24 cookies?

**b.**Louisa only has 10 eggs. She has plenty of other ingredients. How many cookies can she make?

**Ingredients for 12 cookies**

250g flour

200g chocolate chips

125g butter

125g sugar

3 eggs

**a. **Since 24 is a multiple of 12, we donβt 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 Γ 2 = **500g**

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

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

3 eggs = 12 cookies

1 egg = 12 Γ· 3

= 4 cookies

10 eggs = 4 Γ 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.