The short answer
- To add or subtract, put both fractions over a common denominator first.
- To multiply, multiply the tops and the bottoms.
- To divide, flip the second fraction and multiply.
- Simplify by dividing top and bottom by their greatest common divisor.
The words for the parts
In 3/4, the 3 on top is the numerator and the 4 underneath is the denominator. The denominator says how many equal parts the whole is split into; the numerator says how many of those parts you have. A proper fraction is less than 1 (3/4). An improper fraction is 1 or more (7/4). A mixed number is a whole number and a proper fraction (1 3/4).
Simplifying a fraction
A fraction is in its lowest terms when the top and bottom share no factor except 1. Divide both by their greatest common divisor (GCD), found with Euclid’s method: keep dividing and taking the remainder until it is 0.
- 126 ÷ 84 = 1 remainder 42
- 84 ÷ 42 = 2 remainder 0GCD 42
- 84 ÷ 42 and 126 ÷ 422/3
Adding fractions
- Lowest common denominator of 4 and 612
- 1/4 = 3/12 and 1/6 = 2/12
- 3 + 25/12
The lowest common denominator is the lowest common multiple of the two denominators. Multiplying the denominators together (24 here) also works, but leaves more to simplify at the end.
Subtracting fractions
- 3/4 = 9/12 and 1/6 = 2/12
- 9 − 27/12
Multiplying fractions
- (2 × 3) / (3 × 4)6/12
- GCD of 6 and 12 is 61/2
No common denominator is needed. “Of” means multiply: half of three quarters is 1/2 × 3/4 = 3/8.
Dividing fractions
- Flip 1/2 to 2/1 and multiply3/4 × 2/1
- (3 × 2) / (4 × 1)6/4
- Divide by 23/2
It makes sense: there are one and a half halves in three quarters.
Mixed numbers
Turn mixed numbers into improper fractions first: multiply the whole number by the denominator and add the numerator.
- 1 1/2 = 3/2 and 2 2/3 = 8/3
- Over 6: 9/6 + 16/625/6
- 25 ÷ 6 = 4 remainder 14 1/6
Multiplying works the same way: 2 1/4 × 1 1/3 = 9/4 × 4/3 = 36/12 = 3.
Fractions to decimals
Divide the numerator by the denominator. If the simplified denominator has no prime factors other than 2 and 5, the decimal ends (3/8 = 0.375). Otherwise it repeats for ever: 1/3 = 0.333…, written 0.(3), and 5/12 = 0.41666…, written 0.41(6).
Decimals to fractions
Write the decimal over 10, 100 or 1,000 (one zero per decimal place) and simplify. 0.375 = 375/1000 = 3/8, and 0.6 = 6/10 = 3/5. For a rounded decimal such as 0.3333, set a largest denominator under More options and the calculator finds the closest simple fraction: 1/3. The same method finds 355/113 for 3.14159, a famous approximation of pi.
Common fractions table
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.(3) | 33.33% |
| 2/3 | 0.(6) | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1(6) | 16.67% |
| 1/7 | 0.(142857) | 14.29% |
| 1/8 | 0.125 | 12.5% |
| 1/9 | 0.(1) | 11.11% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.08(3) | 8.33% |
Fractions and percentages
A percentage is a fraction out of 100. Multiply the decimal by 100: 3/8 = 0.375 = 37.5%. To go back, put the percentage over 100 and simplify: 35% = 35/100 = 7/20. The percentage calculator handles increases, decreases and reverse percentages, and the discount calculator turns offers like “a third off” into prices.
Common mistakes
Adding the denominators
1/4 + 1/6 is not 2/10. Only the numerators are added, once the denominators match.
- Flipping the first fraction instead of the second when dividing.
- Forgetting to turn mixed numbers into improper fractions before multiplying.
- Leaving the answer unsimplified, such as 6/12 instead of 1/2.
- Rounding a repeating decimal too early, which builds up error.
Fractions in everyday life
Recipes (scale 3/4 cup by 1 1/2 for a bigger batch: 1 1/8 cups), measurements in inches, sharing a bill, and splitting time all use fractions. A quarter of an hour is 15 minutes; the hours and time calculator adds up times for you. For marks and grades, an average of fractions is easier as decimals: see the average calculator.
Comparing fractions
To see which of two fractions is bigger, cross-multiply: multiply each numerator by the other fraction’s denominator. For 3/5 and 5/8, 3 × 8 = 24 and 5 × 5 = 25, so 5/8 is bigger. The calculator confirms it: 3/5 − 5/8 = −1/40, a small negative number. Turning both into decimals (0.6 and 0.625) works too.
Fractions of amounts
To find a fraction of an amount, divide by the denominator and multiply by the numerator. 3/8 of 240 is 240 ÷ 8 = 30, then 30 × 3 = 90. In the calculator, type 3/8, choose ×, and type 240 as the second number. Dividing first keeps the numbers small, which helps with mental maths.