Adding & subtracting fractions
Adding & subtracting: related denominators
Fifths and tenths are different-sized pieces, so you cannot add them as they are. Change one so both match.
Why you cannot add them directly
1/5 + 3/10 cannot be added straight away — fifths and tenths are different sizes. It is like adding 2 metres and 30 centimetres: you must convert to the same unit first. So change one fraction so that both have the SAME denominator.
Making equivalent fractions
Multiply the top AND the bottom by the same number, and the value does not change. 1/5 = 2/10 (multiply both by 2) 1/2 = 5/10 (multiply both by 5) 1/4 = 3/12 (multiply both by 3) The fraction looks different but is worth exactly the same.
Choosing the denominator to use
When one denominator divides into the other, use the BIGGER one. For fifths and tenths, use tenths — because 5 goes into 10. For quarters and twelfths, use twelfths. 1/5 + 3/10 = 2/10 + 3/10 = 5/10 = 1/2
Then it is the same as before
Once the denominators match, the rule is the one you already know: add or subtract the numerators, keep the denominator, then simplify. The only extra step is making them match in the first place. Do that carefully and the rest is easy.
Worked examples
1/5 + 3/10
- 5 goes into 10, so use tenths.
- 1/5 = 2/10.
- 2/10 + 3/10 = 5/10, which simplifies to 1/2.
Answer: 1/5 + 3/10 = 1/2
1/2 + 1/4
- 2 goes into 4, so use quarters.
- 1/2 = 2/4.
- 2/4 + 1/4 = 3/4.
Answer: 1/2 + 1/4 = 3/4
3/4 − 1/8
- 4 goes into 8, so use eighths.
- 3/4 = 6/8.
- 6/8 − 1/8 = 5/8.
Answer: 3/4 − 1/8 = 5/8
Writing 1/5 as tenths
- To turn 5 into 10, multiply by 2.
- Do the same to the top: 1 × 2 = 2.
Answer: 1/5 = 2/10
Finding a missing numerator
- ▢/20 + 1/10 = 3/20.
- Change 1/10 into twentieths: 2/20.
- The numerators must add to 3, so ▢ + 2 = 3.
Answer: The missing numerator is 1
Where you see this in real life
- A recipe asking for 1/2 a cup and 1/4 of a cup — convert to quarters to find the total.
- Adding 1/4 hour and 1/2 hour of work by turning both into quarter-hours.
- Combining 3/10 of a tank and 1/5 of a tank of fuel.
Key words
- Equivalent fraction
- A fraction of equal value written differently.
- Common denominator
- The same bottom number for two or more fractions.
- Numerator
- The top number of a fraction.
- Denominator
- The bottom number of a fraction.
- Simplify
- Cancel a fraction to its smallest form.
- Convert
- Change into a different but equal form.