The rules
a/b + c/d = (a·d + c·b) / (b·d) a/b − c/d = (a·d − c·b) / (b·d) a/b × c/d = (a·c) / (b·d) a/b ÷ c/d = (a·d) / (b·c)
For addition and subtraction the calculator uses the least common denominator (the least common multiple of b and d) rather than b·d, which keeps the numbers small. For example, 1/4 + 2/6 uses 12: 3/12 + 4/12 = 7/12.
How to add fractions by hand
Take 2/3 + 3/4:
- Find the least common denominator of 3 and 4. The smallest number both divide into is 12.
- Rewrite each fraction over 12: 2/3 = 8/12 (multiply top and bottom by 4) and 3/4 = 9/12 (multiply by 3).
- Add the numerators and keep the denominator: 8/12 + 9/12 = 17/12.
- Simplify. 17 and 12 share no common factor, so 17/12 is already in lowest terms.
- Convert to a mixed number: 17 ÷ 12 = 1 remainder 5, so 1 5/12. As a decimal it is about 1.4167.
Subtraction works the same way: 5/6 − 1/4 = 10/12 − 3/12 = 7/12.
Multiplying and dividing
To multiply, multiply straight across: 3/4 × 2/9 = 6/36, which simplifies to 1/6. No common denominator is needed.
To divide, flip the second fraction and multiply: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Flipping a fraction gives its reciprocal, and dividing by a number is the same as multiplying by its reciprocal.
Simplifying and mixed numbers
A fraction is simplified by dividing the numerator and denominator by their greatest common divisor. 6/8 becomes 3/4 because both numbers divide by 2.
An improper fraction becomes a mixed number by dividing the numerator by the denominator: 15/8 = 1 remainder 7, so 1 7/8.
Common fractions as decimals and percents
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.3% |
| 2/3 | 0.666… | 66.7% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 1/10 | 0.1 | 10% |
Common mistakes
- Adding the denominators. 1/2 + 1/3 is not 2/5. Rewrite both over a common denominator first: 3/6 + 2/6 = 5/6.
- Flipping the wrong fraction when dividing. Only the second fraction (the divisor) is flipped.
- Using a mixed number as if it were a fraction. Convert 2 1/3 to 7/3 before multiplying or dividing.