Each operation has one rule. The rule fits in a line, so it comes first. Everything after it is for the moment the rule stops making sense, which is usually around 7pm on a school night with a worksheet on the table.
The pattern worth noticing: adding and subtracting need a common denominator, multiplying and dividing never do. Most fraction mistakes come from mixing those two families up.
Adding fractions
Same denominator. Add the tops, keep the bottom. 2/7 + 3/7 = 5/7. The denominator tells you the size of the pieces, and adding sevenths to sevenths still gives sevenths.
Different denominators. The pieces are different sizes, so you cannot count them together yet. Change both fractions into pieces of the same size first.
- Find a number both denominators divide into. For 1/3 + 1/4, that is 12.
- Convert each fraction. 1/3 becomes 4/12 and 1/4 becomes 3/12.
- Add the numerators. 4/12 + 3/12 = 7/12.
- Simplify if you can. 7/12 is already in its lowest terms.
Mixed numbers. Turn them into improper fractions first. 1 2/3 + 2 1/2 becomes 5/3 + 5/2, which becomes 10/6 + 15/6 = 25/6, which is 4 1/6.
Subtracting works the same way. 3/4 − 1/6: the common denominator is 12, so 9/12 − 2/12 = 7/12.
Multiplying fractions
Multiply straight across. 2/3 × 3/5 = 6/15, which simplifies to 2/5.
No common denominator is needed, and this is where children who have just mastered adding go wrong. When both fractions share a denominator, you still multiply it: 2/5 × 3/5 = 6/25. Keeping the 5 and writing 6/5 is the single most common multiplication error.
Whole numbers. Write the whole number over 1. 3/8 × 4 = 3/8 × 4/1 = 12/8 = 1 1/2.
Mixed numbers. Convert first, every time. 1 1/2 × 2 2/3 becomes 3/2 × 8/3 = 24/6 = 4. Multiplying the whole parts and the fraction parts separately gives 2 1/3, which is wrong.
Cancelling before you multiply. For 4/9 × 3/8, divide the 4 and the 8 by 4, and the 3 and the 9 by 3. That leaves 1/3 × 1/2 = 1/6. Same answer as multiplying out 12/72, with smaller numbers.
Dividing fractions: tip and times
South African teachers call it tip and times. Tip the second fraction upside down, then times. In the UK and the US the same method goes by keep, change, flip: keep the first fraction, change divide to multiply, flip the second.
3/4 ÷ 2/5 → 3/4 × 5/2 = 15/8 = 1 7/8.
The why, for when a child asks. Dividing asks how many of something fit into something else. How many halves fit into 1? Two. So 1 ÷ 1/2 = 2, and 1 × 2/1 gives the same answer. Flipping the second fraction turns "how many of these fit" into a multiplication that answers it.
Two cases that trip children up:
- Dividing by a whole number. Write it over 1 and flip it. 2/3 ÷ 4 = 2/3 × 1/4 = 2/12 = 1/6.
- Mixed numbers. Convert both first. 2 1/4 ÷ 1 1/2 = 9/4 ÷ 3/2 = 9/4 × 2/3 = 18/12 = 1 1/2.
Only the second fraction flips. Flipping the first, or both, is the most common division error.
Finding a fraction of a number
Divide by the denominator, then multiply by the numerator.
3/4 of 20: 20 ÷ 4 = 5, then 5 × 3 = 15.
This is the same as multiplying: 3/4 × 20/1 = 60/4 = 15. Dividing first keeps the numbers small, which is why most teachers teach it that way.
Comparing fractions
To see which of two fractions is bigger, give them the same denominator. 2/3 or 1/2? Converted to sixths, that is 4/6 against 3/6, so 2/3 is bigger.
The shortcut is cross-multiplying. Multiply each numerator by the other fraction's denominator and compare. For 5/8 and 3/5: 5 × 5 = 25 and 3 × 8 = 24. 25 is bigger, so 5/8 is bigger.
A bigger denominator means smaller pieces. 1/8 is smaller than 1/4, because a pizza cut into eight gives smaller slices than one cut into four. Children who are new to fractions often assume the opposite.
When a child keeps getting fractions wrong, it is usually one of these six repeated across every question rather than six different problems. Find which one, fix that, and the marks come back together.
Practice questions
- 1/5 + 2/5
- 1/2 + 1/3
- 5/6 − 1/4
- 3/7 × 2/9
- 2/5 of 35
- 5/6 ÷ 1/3
- 1 3/4 + 2 2/3
- Which is bigger, 3/4 or 5/7?
Answers
- 3/5
- 5/6 (3/6 + 2/6)
- 7/12 (10/12 − 3/12)
- 2/21 (6/63, simplified)
- 14 (35 ÷ 5 = 7, then 7 × 2)
- 2 1/2 (5/6 × 3 = 15/6)
- 4 5/12 (7/4 + 8/3 = 21/12 + 32/12 = 53/12)
- 3/4 (cross-multiply: 3 × 7 = 21, 5 × 4 = 20)
Frequently asked questions
How do you add fractions? Make the denominators the same, then add the numerators and keep the denominator. For 1/3 + 1/4, both become twelfths: 4/12 + 3/12 = 7/12. If the denominators already match, add the numerators straight away.
How do you multiply fractions? Multiply the numerators together and the denominators together. 2/3 × 3/5 = 6/15, which simplifies to 2/5. You do not need a common denominator, and when the denominators match you still multiply them: 2/5 × 3/5 = 6/25.
How do you divide fractions? Flip the second fraction upside down, then multiply. 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8, which is 1 7/8. Only the second fraction flips.
What is tip and times? Tip and times is the name South African teachers use for dividing fractions: tip the second fraction upside down, then times, meaning multiply. In the UK and the US the same method is called keep, change, flip.
Why do you flip the fraction when dividing? Dividing asks how many of one thing fit into another. Half fits into 1 twice, so 1 ÷ 1/2 = 2, which is the same as 1 × 2/1. Flipping the second fraction turns the division into a multiplication that gives the same answer.
When multiplying fractions with the same denominator, do you multiply the denominator? Yes. 2/5 × 3/5 = 6/25. Keeping the denominator as 5 is a common mistake that comes from mixing up the rule for adding, where the denominator does stay the same.
How do you find a fraction of a number? Divide the number by the denominator, then multiply by the numerator. For 3/4 of 20, 20 ÷ 4 = 5, then 5 × 3 = 15.
Is 2/3 more than 1/2? Yes. Converted to sixths, 2/3 is 4/6 and 1/2 is 3/6. Cross-multiplying gives the same result: 2 × 2 = 4 and 1 × 3 = 3, so 2/3 is bigger.
How do you multiply mixed numbers? Convert each mixed number to an improper fraction first, then multiply across. 1 1/2 × 2 2/3 becomes 3/2 × 8/3 = 24/6 = 4. Multiplying the whole numbers and fractions separately gives a wrong answer.
Maths at CambriLearn
CambriLearn is an online school for students aged 4 to 19, teaching live timetabled lessons with specialist teachers across six curriculum pathways: the International British Curriculum*, Pearson Edexcel, US K-12, CAPS, KABV and the IEB. Every lesson is recorded, so a student who missed the fractions lesson, or did not follow it the first time, can watch it again before the test. We are accredited by Cognia and by Pearson Edexcel, registered with SACAI and with the IEB, and NCAA approved. Families have rated us 4.8 stars across Google, Trustpilot, HelloPeter and Facebook.
