A fraction is a part of a whole. In 3/4, the bottom number (the denominator) says the whole is cut into 4 equal parts, and the top number (the numerator) says you have 3 of them. Fractions show up again in algebra, in ratios and in every formula with division, so fixing any shaky spots now pays off later.
Equivalent fractions and simplifying
Multiplying or dividing the top and bottom by the same number doesn’t change a fraction’s value. 1/2 = 2/4 = 50/100.
To simplify, divide the top and bottom by their greatest common factor (GCF).
Example Simplify 18/24
The GCF of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18/24 = 3/4.
Adding and subtracting
You can only add or subtract fractions with the same denominator. Rewrite them over a common denominator first. The least common denominator (LCD) keeps the numbers smallest.
Example Add 2/3 + 3/4
- The LCD of 3 and 4 is 12.
- Rewrite: 2/3 = 8/12 and 3/4 = 9/12.
- Add the numerators: 8/12 + 9/12 = 17/12.
- As a mixed number, 17/12 = 1 5/12.
Example Subtract 5/6 − 1/4
The LCD is 12: 10/12 − 3/12 = 7/12.
Never add the denominators
1/2 + 1/3 is not 2/5. Half a pizza plus a third of a pizza is more than half a pizza, and 2/5 is less. The right answer is 3/6 + 2/6 = 5/6.
Mixed numbers
Convert mixed numbers to improper fractions before you calculate. Multiply the whole number by the denominator and add the numerator.
Example Add 2 1/3 + 1 3/4
- Convert: 2 1/3 = 7/3 and 1 3/4 = 7/4.
- Common denominator 12: 28/12 + 21/12 = 49/12.
- Convert back: 49/12 = 4 1/12.
Multiplying
Multiply straight across: top times top, bottom times bottom. No common denominator needed.
Example Multiply 3/5 × 10/9
Straight across: 30/45 = 2/3.
Simplify before multiplying to keep the numbers small. In 3/5 × 10/9, the 3 and 9 share a factor of 3, and the 5 and 10 share a factor of 5. Cancel them and you get 1/1 × 2/3 = 2/3 directly.
Dividing
To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, and flip the second.
Example Divide 3/4 ÷ 9/8
3/4 × 8/9 = 24/36 = 2/3.
Example Divide 2 1/2 ÷ 1/4
5/2 × 4/1 = 20/2 = 10. That makes sense: there are 10 quarters in two and a half.
Comparing fractions
To compare two fractions, cross-multiply, or convert both to decimals.
Example Which is larger, 5/8 or 3/5?
Cross-multiply: 5 × 5 = 25 and 8 × 3 = 24. Since 25 > 24, 5/8 is larger.
Fractions, decimals and percents
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
Fractions with variables
Algebraic fractions follow exactly the same rules.
Example Add 1/x + 1/(2x)
The LCD is 2x: 2/(2x) + 1/(2x) = 3/(2x).
Try these
Add 3/8 + 5/12.
Show answer
19/24. The LCD is 24: 9/24 + 10/24.
Subtract 7/10 − 2/5.
Show answer
3/10. 2/5 = 4/10.
Multiply 4/9 × 3/8.
Show answer
1/6. 12/72 simplifies to 1/6.
Divide 5/6 ÷ 10/3.
Show answer
1/4. 5/6 × 3/10 = 15/60.
Subtract 3 1/2 − 1 2/3.
Show answer
1 5/6. 7/2 − 5/3 = 21/6 − 10/6 = 11/6.
A recipe needs 2/3 cup of sugar. How much do you need for 1½ batches?
Show answer
1 cup. 2/3 × 3/2 = 1.
Put in order from smallest to largest: 2/3, 5/8, 7/10.
Show answer
5/8, 2/3, 7/10. As decimals they are about 0.625, 0.667 and 0.7.
Practice
Questions students ask
Do I always have to simplify?
Most teachers and tests expect fractions in lowest terms. On the SAT, a student-produced answer can be entered as an equivalent fraction or decimal, but simplifying first avoids typing mistakes.
Is an improper fraction wrong?
No. 17/12 is a perfectly good answer and often easier to use in the next step of a problem. Convert to a mixed number only when asked.
How do fractions connect to percentages?
A percent is a fraction with a denominator of 100. See Percentages for percent change, discounts and interest.