Fractions confuse more students than almost any other topic in school maths – yet the rules are simple once you see them laid out. This guide explains how to add fractions with the same and different denominators, how to subtract, multiply and divide fractions, how to work with mixed numbers, how to simplify fractions, and how to convert between fractions, decimals and percentages – all with worked examples.
Parts of a Fraction
A fraction such as 3/4 has two parts. The numerator (top number, 3) says how many parts you have. The denominator (bottom number, 4) says how many equal parts the whole is divided into. So 3/4 means three of four equal parts.
- Proper fraction: numerator smaller than denominator, such as 3/4.
- Improper fraction: numerator equal to or larger than the denominator, such as 7/4.
- Mixed number: a whole number and a fraction, such as 1 3/4 (which equals 7/4).
How to Simplify Fractions
To simplify a fraction (reduce it to lowest terms), divide the numerator and denominator by their greatest common divisor (GCD), also called the highest common factor (HCF).
Example: simplify 18/24. The GCD of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4.
If you do not spot the GCD, divide by any common factor (such as 2 or 3) repeatedly until no common factor remains. Always simplify your final answer.
Adding Fractions with the Same Denominator
When the denominators are equal, add the numerators and keep the denominator:
a/c + b/c = (a + b)/c
Example: 2/7 + 3/7 = 5/7.
How to Add Fractions with Different Denominators
To add fractions with different denominators, first rewrite them with a common denominator – ideally the least common multiple (LCM) of the denominators.
Example: 1/2 + 1/3
- The LCM of 2 and 3 is 6.
- Convert: 1/2 = 3/6 and 1/3 = 2/6.
- Add the numerators: 3/6 + 2/6 = 5/6.
A quick shortcut for two fractions is the cross-multiplication rule: a/b + c/d = (ad + bc)/bd, then simplify. For 1/2 + 1/3: (1×3 + 1×2)/(2×3) = 5/6.
How to Subtract Fractions
Subtraction works the same way: find a common denominator, then subtract the numerators.
Example: 3/4 − 2/5. The LCM of 4 and 5 is 20. 3/4 = 15/20 and 2/5 = 8/20, so 15/20 − 8/20 = 7/20.
How to Multiply Fractions
Multiplying fractions is the easiest operation – no common denominator needed. Multiply the numerators together and the denominators together:
a/b × c/d = (a × c)/(b × d)
Example: 2/3 × 3/4 = 6/12 = 1/2. You can also cancel common factors before multiplying: the 3s cancel, giving 2/4 = 1/2.
To find a fraction of a number, multiply: 3/5 of 40 = 3/5 × 40 = 24.
How to Divide Fractions
To divide fractions, keep the first fraction, change ÷ to ×, and flip the second fraction (use its reciprocal) – often remembered as "keep, change, flip":
a/b ÷ c/d = a/b × d/c
Example: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10.
Working with Mixed Numbers
Convert mixed numbers to improper fractions first: multiply the whole number by the denominator and add the numerator. For example, 2 1/3 = (2 × 3 + 1)/3 = 7/3.
Example: 2 1/3 + 1 3/4
- Convert: 7/3 + 7/4
- Common denominator 12: 28/12 + 21/12 = 49/12
- Convert back: 49 ÷ 12 = 4 remainder 1, so the answer is 4 1/12.
Comparing Fractions
To see which fraction is larger, give them a common denominator or cross-multiply. Is 5/8 bigger than 3/5? Cross-multiply: 5 × 5 = 25 and 3 × 8 = 24. Since 25 > 24, 5/8 is larger. Converting both to decimals (0.625 and 0.6) gives the same answer.
Fractions, Decimals and Percentages
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 3/8 | 0.375 | 37.5% |
| 2/3 | 0.667 | 66.67% |
- Fraction to decimal: divide the numerator by the denominator (3 ÷ 8 = 0.375).
- Decimal to fraction: write the decimal over a power of 10 and simplify (0.375 = 375/1000 = 3/8).
- Fraction to percentage: multiply the decimal by 100 (0.375 = 37.5%). See our guide on how to calculate percentage.
Fractions in Everyday Life
- Cooking: halving a recipe that needs 3/4 cup of flour – 3/4 × 1/2 = 3/8 cup.
- Money: splitting a bill – each of 4 friends pays 1/4.
- Time: 45 minutes is 3/4 of an hour.
- Measurements: carpentry and tailoring often use inches in halves, quarters and eighths.
- Discounts: "one-third off" means paying 2/3 of the price.
Common Fraction Mistakes
- Adding denominators: 1/2 + 1/3 is not 2/5.
- Forgetting to flip the second fraction when dividing.
- Not converting mixed numbers before multiplying or dividing.
- Leaving answers unsimplified.
- Cancelling terms that are added rather than multiplied.
How to Find the Least Common Denominator
The least common denominator is the smallest number that all the denominators divide into – the LCM of the denominators. Two quick methods:
- List multiples: for 4 and 6, multiples of 4 are 4, 8, 12, 16… and of 6 are 6, 12, 18… The first shared one is 12.
- Prime factors: 4 = 2 × 2 and 6 = 2 × 3. Take each prime the most times it appears: 2 × 2 × 3 = 12.
Using the least common denominator keeps the numbers small, so there is less simplifying at the end. Multiplying the denominators together (4 × 6 = 24) also works, but you will need to simplify more afterwards.
Adding Three or More Fractions
Find one common denominator for all of them. Example: 1/2 + 2/3 + 3/4. The LCM of 2, 3 and 4 is 12, so the sum is 6/12 + 8/12 + 9/12 = 23/12 = 1 11/12.
Fraction Word Problems
- Pizza: Riya eats 3/8 of a pizza and Aman eats 1/4. How much is left? 3/8 + 2/8 = 5/8 eaten, so 3/8 is left.
- Salary: Priya spends 2/5 of her salary on rent and 1/4 on food. What fraction remains? 8/20 + 5/20 = 13/20 spent, so 7/20 remains.
- Recipe: a recipe needs 2 1/2 cups of rice for 5 people. How much for 8 people? 5/2 ÷ 5 × 8 = 4 cups.
- Distance: a runner covers 3/4 km in 5 minutes. How far in 1 hour? 3/4 × 12 = 9 km.
Fractions with Negative Numbers
The usual sign rules apply. A negative fraction can be written with the minus in front or in the numerator: −3/4 = (−3)/4. When adding, find the common denominator first, then add the signed numerators: 1/3 − 3/4 = 4/12 − 9/12 = −5/12. When multiplying or dividing, a negative times a positive is negative and two negatives make a positive.
A Short History of Fractions
Fractions are thousands of years old. Ancient Egyptians wrote most fractions as sums of unit fractions such as 1/2 + 1/4, and Indian mathematicians wrote fractions with one number above another – the ancestor of today's notation. The horizontal fraction bar became common in Europe through Arabic mathematics in the Middle Ages.
Frequently Asked Questions
How do you add fractions with different denominators?
Find a common denominator, ideally the least common multiple, convert each fraction, then add the numerators. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
How do you multiply fractions?
Multiply the numerators together and the denominators together, then simplify. 2/3 × 3/4 = 6/12 = 1/2.
How do you divide fractions?
Keep the first fraction, change division to multiplication and flip the second fraction. 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10.
How do you simplify a fraction?
Divide the numerator and denominator by their greatest common divisor. 18/24 ÷ 6/6 = 3/4.
How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator and add the numerator, keeping the same denominator. 2 1/3 = 7/3.
How do you convert a fraction to a decimal?
Divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375.
Which is bigger, 5/8 or 3/5?
5/8, because 5/8 = 0.625 and 3/5 = 0.6. Cross-multiplying gives 25 versus 24.
How do you find a common denominator?
Find the least common multiple of the denominators. For 4 and 6 it is 12, so 1/4 = 3/12 and 1/6 = 2/12.
How do you add three fractions?
Convert all of them to one common denominator and add the numerators. 1/2 + 2/3 + 3/4 = 6/12 + 8/12 + 9/12 = 23/12.
Note: Answers are given in simplest form. Use the fraction calculator to check homework, but try each problem by hand first to learn the method.