# Solve Equations with Fractions

In this worksheet, students will learn how to solve linear equations involving fractions, to find the value of a variable represented by a letter.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:

### QUESTION 1 of 10

"I think of a number, divide by 5, then add 4 and the answer is 7. What number am I thinking of?"

From your previous work on equations, you will be familiar with this sort of puzzle and how to express it algebraically.

Well, this includes division, which is important when we think about how to write it as an equation.

When writing algebra we do not use the '÷' symbol, as we write a division sum as a fraction

So in order to write this puzzle as an equation, with x as the unknown number, we need to write:

Other than this, there is no difference between this and other two-step equations.

In this case, x has been divided by 5 then had 4 added.

To solve this, we must apply the inverse operations in the reverse order.

So first we subtract 4 then we multiply by 5, remembering to do the same to both sides.

We can set out our working as like this:

So the number we started with was 15.

Here is another equation, which is slightly different:

Notice here that the line goes under the x - 12, not just the x.

This changes the order, meaning that starting with x first, we subtract 12 then divide by 3.

So when we apply the inverses in reverse order, we must multiply by 3 then add 12.

Here is our working:

Are you getting the hang of this?

In this activity, we will solve equations involving fractions to find the value of a variable represented by a letter.

You will want to have a piece of paper and pen handy to record your working, as this is a great way to maximise your marks in an exam.

Choose the correct solution to the equation:

x = -1

x = 10

x = 13

x = 17

Here's an equation involving a different letter representing the unknown element:

Which of the options below shows the correct solution?

n = 3

n = 8

n = 33

n = 48

Can you solve this equation?

n = 3

n = 8

n = 33

n = 48

Solve this equation:

n = 3

n = 8

n = 33

n = 48

Can you match the following equations to their solutions?

## Column B

x/7 = 5
x = 35
x/3 = 8
x = 32
x/4 - 3 = 5
x = 28
x/4 - 5 = 2
x = 24

Can you match each equation below to its solution?

## Column B

4y/5 + 6 = 18
y = 15
3y/4 - 1 = 8
y = 5
7y/3 + 5 = 12
y = 12
8y/5 + 3 = 11
y = 3

x = 8 is the solution to at least one of the equations below.

Which of the equations below does this solution work for?

(2x - 1)/3 = 5

(5x + 2)/2 = 23

(4x + 3)/5 = 7

(8x - 9)/5 = 10

x = -20 is the solution to at least one of the equations below.

Which of the equations below does this solution work for?

x/4 = -5

(x - 4)/8 = -4

x/5 + 9 = 5

(3x + 24)/2 = -10

Can you match the equations below to their solutions?

Solve the equation below.

• Question 1

Choose the correct solution to the equation:

x = 10
EDDIE SAYS
x has been divided by 2 then had 7 added to this answer. So to solve this, we need to subtract 7 then multiply by 2: x/2 + 7 = 12 x/2 + 7 - 7 = 12 - 7 x/2 = 5 x/2 × 2 = 5 × 2 x = 10 How did you get on with this first one? It's a good idea to review the Introduction to make sure you are confident with this method before moving on to tackle the rest of the questions in this activity.
• Question 2

Here's an equation involving a different letter representing the unknown element:

Which of the options below shows the correct solution?

n = 48
EDDIE SAYS
How did you do? Remember that the letter itself is not important, it just stands for a value we don't know. The inverse of ÷ 4 is x 4 and the inverse of - 5 is + 5. So we find our solution by following these steps: n/4 - 5 = 7 n/4 - 5 + 5 = 7 + 5 n/4 = 12 n/4 × 4 = 12 × 4 n = 48
• Question 3

Can you solve this equation?

EDDIE SAYS
To solve this we need to × 3 then - 17: (P + 17) / 3 = 8 ((P + 17) / 3) × 3 = 8 × 3 P + 17 = 24 P + 17 - 17 = 24 - 17 P = 7
• Question 4

Solve this equation:

EDDIE SAYS
The numbers are a bit bigger here, but our method needs to be just the same. (s - 38) / 11 = 12 ((s - 38) / 11) × 11 = 12 × 12 s - 38 = 132 s - 38 + 38 = 132 + 38 s = 170
• Question 5

Can you match the following equations to their solutions?

## Column B

x/7 = 5
x = 35
x/3 = 8
x = 24
x/4 - 3 = 5
x = 32
x/4 - 5 = 2
x = 28
EDDIE SAYS
There is a lot to work through here and our working could take up a lot of space, so we'll keep it brief. Here are the inverses in the correct order for each equation: x/7 = 5 → × 7 → x = 35 x/3 = 8 → × 3 → x = 24 x/4 - 3 = 5 → + 3 then × 4 → x = 32 x/4 - 5 = 2 → + 5 then × 4 → x = 28 Did you get them all sorted and matched correctly?
• Question 6

Can you match each equation below to its solution?

## Column B

4y/5 + 6 = 18
y = 15
3y/4 - 1 = 8
y = 12
7y/3 + 5 = 12
y = 3
8y/5 + 3 = 11
y = 5
EDDIE SAYS
Did you spot that these are actually 3-step equations? Here are our shortened solutions: 4y/5 + 6 = 18 → - 6 then × 5 then ÷4 → y = 15 3y/4 - 1 = 8 → + 1 then × 4 then ÷ 3 → y = 12 7y/3 + 5 = 12 → - 5 then × 3 then ÷ 7 → y = 3 8y/5 + 3 = 11 → - 3 then × 5 then ÷ 8 → y = 5 How are you doing? Don't worry if you make a mistake, so long as you can see why you went wrong. Remember that marks are available for your working too, so make sure you are writing these out on your paper.
• Question 7

x = 8 is the solution to at least one of the equations below.

Which of the equations below does this solution work for?

(2x - 1)/3 = 5
(4x + 3)/5 = 7
EDDIE SAYS
These equations have 3 steps to them as well. We can check each equation by replacing x with 8 and seeing if it gives the right answer: (2x - 1)/3 = 5 → (2 × 8 - 1)/3 = 15 ÷ 3 = 5 (correct) (5x + 2)/2 = 23 → (5 × 8 + 2)/2 = 42 ÷ 2 = 21 (wrong) (4x + 3)/5 = 7 → (4 × 8 + 3)/5 = 35 ÷ 5 = 7 (correct) (8x - 9)/5 = 10 → (8 × 8 - 9)/5 = 55 ÷ 5 = 11 (wrong) So two of these equations worked with this solution. Did you spot them both?
• Question 8

x = -20 is the solution to at least one of the equations below.

Which of the equations below does this solution work for?

x/4 = -5
x/5 + 9 = 5
EDDIE SAYS
Remember, we can check each equation by replacing x with -20. x/4 = -5 → -20 ÷ 4 = -5 (correct) (x - 4)/8 = -4 → (-20 - 4)/8 = -24 ÷ 8 = -3 (wrong) x/5 + 9 = 5 → -20 ÷ 5 + 9 = -4 + 9 = 5 (correct) (3x + 24)/2 = -10 → (3 x -20 + 24)/2 = -36 ÷ 2 = -18 (wrong) So this time there were 2 equations again which matched this solution.
• Question 9

Can you match the equations below to their solutions?

EDDIE SAYS
Let's work out each equation, one at a time, to see which solution matches with each. 2n/3 = 8 → × 3 then ÷ 2 → n = 12 n/2 + 7 = 12 → -7 then × 2 → n = 10 (n - 7)/4 = 1 → × 4 then + 7 → n = 11 (5n + 1)/6 = 11 → × 6 then - 1 then ÷ 5 → n = 13 How did you do with these? Just one more challenge left to complete now.
• Question 10

Solve the equation below.

EDDIE SAYS
This is a tricky one! There are some nasty numbers here, but our method is still the same, so let's just take it one step at a time. 3x/4 + 2 = -5.5 3x/4 + 2 - 2 = -5.5 - 2 3x/4 = -7.5 3x/4 × 4 = -7.5 × 4 3x = -30 3x ÷ 3 = -30 ÷ 3 x = -10 If you got this one right, then you are an equation solving expert! And that's the end of this activity. Well done - give yourself a pat on the back.
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