In this article
Fractions look hard only because there are four different rules for four different jobs. Once you see which job you are doing, each one takes about thirty seconds. This guide takes the four operations in turn with the same style of example each time, then covers simplifying and mixed numbers, which is where most marks are lost.
The four rules at a glance
| Job | Rule | Example | Answer |
|---|---|---|---|
| Add | Common denominator, add tops | 1/2 + 1/3 = 3/6 + 2/6 | 5/6 |
| Subtract | Common denominator, subtract tops | 3/4 − 2/5 = 15/20 − 8/20 | 7/20 |
| Multiply | Tops together, bottoms together | 2/3 × 3/4 = 6/12 | 1/2 |
| Divide | Flip the second, then multiply | 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 | 5/4 = 1 1/4 |
Check the last column against the calculator below. Getting the same answer by hand and by tool is the best way to learn.
Adding and subtracting: the common denominator
You can only add things of the same size. A half and a third are different sizes of slice, so cut both into the same small slices first. Six is a number that both 2 and 3 divide into, so use sixths.
- 1Find a common denominatorThe smallest number that 2 and 3 both go into is 6.
- 2Rewrite each fraction1/2 = 3/6 (multiply top and bottom by 3). 1/3 = 2/6 (multiply by 2).
- 3Add the tops3/6 + 2/6 = 5/6. The bottom stays 6.
- 4Simplify5 and 6 share no factor, so 5/6 is final.
The bottom number tells you the size of the slice. It does not get added.
- 1 + 1 over 2 + 3
- Gives 0.4, which is smaller than 1/2
- Adding something cannot make it smaller
- 3/6 + 2/6
- Gives 0.833…
- Bigger than both 1/2 and 1/3, as it should be
A quick sense-check: the sum of two positive fractions must be larger than each of them.
Subtraction works the same. For 3/4 − 2/5 the smallest common denominator is 20: 3/4 = 15/20 and 2/5 = 8/20. So 15/20 − 8/20 = 7/20.
Multiplying: the easy one
Multiply the tops, multiply the bottoms. 2/3 × 3/4 = 6/12, and dividing top and bottom by 6 gives 1/2. You can also cancel first: the 3 on top and the 3 on the bottom cancel, leaving 2/4, which is 1/2.
Think of it as “a fraction of a fraction”. Three quarters of two thirds of a pizza is half a pizza.
Dividing: flip and multiply
Dividing by 2/3 asks “how many two-thirds fit in this?”. The shortcut is to flip the second fraction (swap top and bottom) and then multiply. 5/6 ÷ 2/3 becomes 5/6 × 3/2 = 15/12, which simplifies to 5/4, or 1 1/4.
Simplifying with the GCD
A fraction is in simplest form when top and bottom share no factor except 1. Divide both by their greatest common divisor (GCD). For 18/24 the GCD is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4. The answer is 3/4.
If you do not spot the GCD, divide by any common factor you can see, such as 2, and repeat until nothing more divides.
Mixed numbers and improper fractions
A mixed number like 2 3/4 has a whole part and a fraction. An improper fraction like 11/4 has a top bigger than its bottom. They are the same quantity. To add mixed numbers, convert to improper fractions first.
- 1Make them improper2 3/4 = (2 × 4 + 3)/4 = 11/4. And 1 5/6 = (1 × 6 + 5)/6 = 11/6.
- 2Common denominator11/4 = 33/12 and 11/6 = 22/12.
- 3Add33/12 + 22/12 = 55/12.
- 4Turn back into a mixed number55 ÷ 12 = 4 remainder 7, so 4 7/12.
To convert back: divide the top by the bottom. The quotient is the whole part and the remainder goes over the same bottom.
Fractions, decimals and percentages
A fraction is just a division. Divide the top by the bottom to get the decimal, then multiply by 100 for a percentage.
1/3 is 0.3333… and never ends, so it is usually rounded. Thirds, sixths and ninths give repeating decimals. Halves, fifths and eighths give tidy ones.
Percent change uses the same idea. See up 50 percent, down 50 percent for why fractions of a changing base surprise people.
Quick checks before you hand in an answer
- Is the bottom number zero anywhere? That is not allowed.
- Did you simplify? 6/12 is correct but not finished.
- Does the size make sense? Adding should make it bigger, multiplying by a fraction below 1 should make it smaller.
- For mixed numbers, did you convert before multiplying or dividing? You cannot multiply the wholes and the fractions separately.
Questions people ask
How do you add fractions with different denominators?
Find a common denominator, rewrite each fraction with it, then add the tops and keep the bottom. For 1/2 + 1/3 use 6: 3/6 + 2/6 = 5/6.
Why is 1/2 + 1/3 not 2/5?
Adding tops and bottoms treats the slices as if they were the same size. 2/5 is 0.4, which is less than 1/2, so it cannot be the sum of 1/2 and something positive. The correct answer is 5/6.
How do you divide fractions?
Flip the second fraction and multiply. 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4.
How do you simplify a fraction?
Divide the top and bottom by their greatest common divisor. 18/24 divided by 6 on each part gives 3/4.
How do you turn an improper fraction into a mixed number?
Divide the top by the bottom. The whole part is the quotient and the remainder goes over the original bottom. 17/5 = 3 remainder 2 = 3 2/5.
How do you convert a fraction to a decimal?
Divide the top by the bottom. 3/8 = 3 ÷ 8 = 0.375. Multiply by 100 for the percentage, 37.5%.
Do I need a common denominator to multiply fractions?
No. Multiply the tops, multiply the bottoms, then simplify. A common denominator is needed only for adding and subtracting.