Fraction arithmetic
Add, subtract, multiply, and divide fractions.
In your head, jot if needed · no calculator why?
Opens at level 12. You're level 1. You can read and practice here now.
the idea
Fractions follow a few fixed rules, and most mistakes come from using one operation's rule for another.
To add or subtract, the pieces must be the same size, so first rewrite both fractions over a common denominator, a bottom number that both denominators divide into. To multiply, multiply the tops and multiply the bottoms. To divide, flip the second fraction and multiply. A fraction of a number is a multiplication: 3/4 of 20 is 3/4 × 20.
Give every answer in lowest terms. An answer bigger than 1, like 5/4, stays a fraction.
techniques
Find one part first
- Divide the number by the bottom of the fraction. That is one part.
- Multiply by the top to get that many parts.
worked example
What is 5/6 of 42?
- One sixth: 42 ÷ 6 = 7.
- Five sixths: 5 × 7 = 35.
Answer: 35
Common denominator first
- Find a number both bottoms divide into. The smallest one keeps the numbers small.
- Rewrite each fraction over it, multiplying its top and bottom by the same number.
- Add or subtract the tops, and keep the bottom.
- Simplify to lowest terms.
worked example
5/6 − 1/4. Answer as a fraction in lowest terms.
- 6 and 4 both divide into 12.
- 5/6 = 10/12, and 1/4 = 3/12.
- 10/12 − 3/12 = 7/12.
Answer: 7/12
Multiply across; flip to divide
- To divide, flip the second fraction upside down and multiply instead.
- Cancel first: divide any top and any bottom by a factor they share.
- Multiply the tops, then the bottoms.
- Check that the answer is in lowest terms.
worked example
3/4 ÷ 9/10. Answer as a fraction in lowest terms.
- Flip and multiply: 3/4 × 10/9.
- Cancel 3 with 9, and 10 with 4: 1/2 × 5/3.
- Tops 1 × 5, bottoms 2 × 3: 5/6.
Answer: 5/6
watch out for
- Adding the tops and adding the bottoms. 2/3 + 1/4 is 11/12, not 3/7, so rewrite both as twelfths first.
- Multiplying without flipping when you divide. 5/6 ÷ 2/3 means 5/6 × 3/2, not 5/6 × 2/3.
- Leaving an answer unsimplified. 6/36 is the right amount, but it must be written as 1/6.
- Finding one part and stopping. 3/8 of 64 is three eighths: 3 × 8, not 8 alone.
practice
Fraction of a number
worked example
What is 1/2 of 152?
Answer: 76
- 1/2 of 152 is 152 ÷ 2 = 76.
Add & subtract
worked example
8/9 − 5/6 Answer as a fraction in lowest terms.
Answer: 1/18
- Use a common denominator of 18: 16/18 − 15/18 = 1/18.
Multiply
worked example
3/4 × 4/5 Answer as a fraction in lowest terms.
Answer: 3/5
- Multiply the tops and the bottoms: 3 × 4 = 12 and 4 × 5 = 20, so 12/20.
- Divide top and bottom by 4: 12 ÷ 4 = 3 and 20 ÷ 4 = 5, so 3/5. Cancelling first is faster.
Divide
worked example
1/6 ÷ 5/6 Answer as a fraction in lowest terms.
Answer: 1/5
- Dividing by 5/6 is multiplying by 6/5: 1/6 × 6/5 = 6/30, which simplifies to 1/5.