How to Solve Equations with Fractions Step by Step

This lesson is for Year 9 and 10 students learning to solve linear equations. It shows you how to solve equations with fractions, starting with one-step equations and building up to an exam-style question.

What does the fraction bar mean in an equation?

Watch this part from 0:00

When you solve for x, your goal is to get x on its own on one side of the equation. Equations with fractions aren’t too difficult. A lot of students see the fraction and panic.

The fraction bar is a division sign. The bottom of the fraction is the number dividing x. So x/5 means x ÷ 5.

Think of x as someone at a party who wants everyone else to go away. Your job is to remove every other number from the side x is on, so that only x is left.

How do you undo an equation step by step?

Watch this part from 1:02

BODMAS gives the order for working out an expression. You may also see it as PEMDAS, which is the same thing. Normally you work through it from left to right: brackets, then indices (or exponents), then division and multiplication, then addition and subtraction.

Solving an equation undoes that work, so you use reverse BODMAS and go from right to left. Here is the method.

1. Work only on the side with x. Leave the other side alone until you need to balance it.
2. Undo addition and subtraction first. Then undo multiplication and division. Indices and brackets come last.
3. Use the opposite operation. The opposite of addition is subtraction. The opposite of division is multiplication.
4. Do the same thing to both sides. Whatever you do to one side of the equals sign, you do to the other, so the equation stays balanced.
5. Check your answer. Substitute it back into the original equation and make sure both sides match.

There are other opposite operations, such as squares and square roots. The equations in this lesson only need the two above.

How do you solve a simple equation with a fraction?

Watch this part from 2:04

Start with x/5 = 4. Nothing is added or subtracted on the side with x, so move straight to multiplication and division.

The 5 is dividing x, so the opposite operation is to multiply by 5. Multiply both sides by 5 to keep the equation balanced. That gives x = 4 × 5, so x = 20.

Check it by substituting x = 20 back in. 20/5 = 4, so the answer is correct.

When a number is subtracted as well

In x/3 − 2 = 4, look for addition or subtraction first. There is a −2, so add 2 to both sides. The −2 and +2 cancel, which leaves x/3 = 6.

Now multiply both sides by 3 to remove the 3 from the bottom. That gives x = 6 × 3, so x = 18.

To check, 18/3 = 6 and 6 − 2 = 4, so the answer is correct.

How do you solve an equation like 3x/4 = 9?

Watch this part from 4:47

The bar still means division, so remove the 4 first. Multiply both sides by 4 to get 3x = 36.

The 3 is multiplying x, so divide both sides by 3. That gives x = 12. To check, 3 × 12 = 36 and 36/4 = 9.

The quicker way: multiply by the reciprocal

You can do both steps at once by multiplying both sides by the reciprocal of 3/4, which is 4/3. This is the same as multiplying by 4 and dividing by 3. It is a little trickier to think about, but it takes one step instead of two.

On the left, 4/3 × 3x/4 = 12x/12, which is just x.

On the right, 9 × 4/3 = 36/3 = 12.

So x = 12, the same answer as before.

Why can’t you subtract from the top of the fraction first?

Watch this part from 6:49

In (x + 7)/3 = 5, you might want to follow reverse BODMAS and subtract 7 first. That is the mistake to avoid.

Written on one line, the left side is (x + 7) ÷ 3. The top of the fraction acts as if it is in brackets. Brackets come first in BODMAS, so they are the last thing you undo.

The rule is that you can’t touch the top of the fraction until the bottom has been removed. Multiply by the denominator first.

1. (x + 7)/3 = 5
2. Multiply both sides by 3: x + 7 = 15
3. Subtract 7 from both sides: x = 8
4. Check: (8 + 7)/3 = 15/3 = 5

The same rule applies to (2x + 3)/5 = 5. Multiply both sides by 5 to get 2x + 3 = 25. Subtract 3 to get 2x = 22, then divide by 2 to get x = 11. To check, 2 × 11 + 3 = 25 and 25/5 = 5.

How do you add fractions in an equation?

Watch this part from 9:12

Adding fractions in an equation is where a lot of students get stuck. It works the same way as adding ordinary fractions. You need a common denominator before you add or subtract.

In x/2 + x/6 = 10, the 2 needs to become a 6. A lot of students say they will multiply by 3. You are really multiplying by 3/3, and 3/3 is 1. Multiplying by 1 keeps the fraction’s value the same.

If you multiplied the whole fraction by 3, the denominator would stay as 2 and you would get 3x/2. That is a different number, so it would change the equation.

Multiplying by 3/3 only changes the way the fraction is written. That is why you don’t need to do anything to the other side of the equals sign.

1. x/2 + x/6 = 10
2. Multiply x/2 by 3/3: 3x/6 + x/6 = 10
3. The denominators match, so add the tops: 4x/6 = 10
4. Simplify if you like: 2x/3 = 10
5. Multiply both sides by 3: 2x = 30
6. Divide both sides by 2: x = 15
7. Check: 15/2 + 15/6 = 45/6 + 15/6 = 60/6 = 10

At step 3 you could also multiply both sides by 6 straight away. Simplifying first is up to you.

What if a number is added outside the fraction?

Watch this part from 11:49

In (x + 4)/2 + 3 = 10, the +3 sits after the fraction, outside it. This time you do subtract first, because that is the normal reverse order of operations.

Subtract 3 from both sides to get (x + 4)/2 = 7. Multiply both sides by 2 to get x + 4 = 14. Subtract 4 from both sides to get x = 10.

To check, (10 + 4)/2 = 7 and 7 + 3 = 10.

How do you solve an equation with fractions on both sides?

Watch this part from 12:40

In (x + 1)/2 = (x − 3)/3, there are x terms on both sides. The end goal is to get the x terms grouped together.

Before anything else, clear the fractions. Multiply both sides by a number that removes both denominators. Multiplying the denominators together gives 2 × 3 = 6.

Don’t expand the brackets straight away. Because the 6 sits outside each bracket, you can write each side as a fraction times the bracket. Those fractions then simplify easily.

1. (x + 1)/2 = (x − 3)/3
2. Multiply both sides by 6: 6(x + 1)/2 = 6(x − 3)/3
3. Write each side as a fraction times the bracket: 6/2 × (x + 1) = 6/3 × (x − 3)
4. Simplify the fractions: 3(x + 1) = 2(x − 3)
5. Expand the brackets: 3x + 3 = 2x − 6
6. Subtract 2x from both sides: x + 3 = −6
7. Subtract 3 from both sides: x = −9

How do you solve a worded problem with fractions?

Watch this part from 15:13

Here is the worded problem. Ben spends a quarter of his weekly pay on rent and $95 on food, which leaves him $205. Find his weekly pay.

Start by defining the variable. Let x be Ben’s weekly pay. Don’t start writing letters in an equation until you have said what each one stands for.

Then build the equation from the words. His weekly pay, minus a quarter of it, minus the $95 for food, equals the $205 he has left. A quarter of his pay is x/4, so the equation is x − x/4 − 95 = 205.

To combine x and x/4, multiply x by 4/4 so it becomes 4x/4. Add 95 to both sides to get 4x/4 − x/4 = 300, which is 3x/4 = 300.

To remove the 3/4, multiply both sides by the reciprocal, 4/3. This is the same as multiplying by 4 and dividing by 3. It leaves x = 1200/3, so x = 400.

Finish with a worded answer, because a worded question needs one: Ben’s weekly pay is $400.

How do you solve a harder exam-style equation with fractions?

Watch this part from 17:38

The last question is exam style, so it takes more steps. The equation is 2(x − 3)/5 = x/2 − 4.

First, get the x terms on one side by subtracting x/2 from both sides. You can also write the 2/5 in front of the bracket. That is personal preference and doesn’t change the answer. Writing x/2 is the same as writing half of x.

Next, the two x terms need a common denominator of 10. Multiply the 2/5 by 2/2 and multiply x/2 by 5/5.

1. 2(x − 3)/5 = x/2 − 4
2. Subtract x/2 from both sides and write 2/5 in front of the bracket: 2/5 × (x − 3) − x/2 = −4
3. Give both fractions a denominator of 10: 4/10 × (x − 3) − 5x/10 = −4
4. Expand the bracket: 4x/10 − 12/10 − 5x/10 = −4
5. Collect the x terms: −x/10 − 12/10 = −4
6. Add 12/10 to both sides, writing −4 as −40/10: −x/10 = −40/10 + 12/10
7. Combine the right side: −x/10 = −28/10
8. Multiply both sides by 10 to clear the fractions: −x = −28
9. Change the sign on both sides: x = 28

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• 55 minVideoHow to Factorise Using the Highest Common Factor (HCF)