Working with fractions
Fractions turn up in recipes, woodworking measurements, homework and probability. Adding 3/4 and 5/6 by hand means finding a common denominator, adding, then simplifying, and each step is a chance for a slip. This calculator does the four operations, reduces the answer to lowest terms and shows it as a mixed number, a decimal and a percentage.
The rules
- Addition: a/b + c/d = (a × d + c × b) ÷ (b × d).
- Subtraction: a/b − c/d = (a × d − c × b) ÷ (b × d).
- Multiplication: a/b × c/d = (a × c) ÷ (b × d).
- Division: a/b ÷ c/d = a/b × d/c = (a × d) ÷ (b × c).
- Simplifying: divide the numerator and the denominator by their greatest common divisor (GCD).
Worked example: 3/4 + 5/6
- Cross multiply: 3 × 6 + 5 × 4 = 18 + 20 = 38.
- Multiply the denominators: 4 × 6 = 24. The raw result is 38/24.
- The GCD of 38 and 24 is 2, so the simplified result is 19/12.
- As a mixed number: 19 ÷ 12 = 1 remainder 7, so 1 7/12.
- As a decimal, 1.5833, or 158.33 percent.
Using the least common denominator, 12, gives the same answer with smaller numbers: 9/12 + 10/12 = 19/12.
3/4 + 5/d for other denominators
| Sum | Result | Decimal |
|---|---|---|
| 3/4 + 5/2 | 13/4 | 3.2500 |
| 3/4 + 5/3 | 29/12 | 2.4167 |
| 3/4 + 5/4 | 2/1 | 2.0000 |
| 3/4 + 5/6 | 19/12 | 1.5833 |
| 3/4 + 5/8 | 11/8 | 1.3750 |
| 3/4 + 5/12 | 7/6 | 1.1667 |
Finding the greatest common divisor
The calculator uses Euclid’s algorithm, more than 2,000 years old: divide the larger number by the smaller, keep the remainder, and repeat until the remainder is zero. For 38 and 24: 38 = 1 × 24 + 14, 24 = 1 × 14 + 10, 14 = 1 × 10 + 4, 10 = 2 × 4 + 2, 4 = 2 × 2 + 0. The last nonzero remainder, 2, is the GCD.
Fractions in everyday measurements
- Tape measures in the US are marked in sixteenths of an inch. Adding 2 3/8 in and 1 5/16 in: convert to sixteenths, 38/16 + 21/16 = 59/16 = 3 11/16 in.
- Recipes: halving 3/4 cup gives 3/8 cup, which is 6 tablespoons.
- Probability: the chance of two heads in a row is 1/2 × 1/2 = 1/4.
To enter a mixed number such as 2 3/8, convert it to an improper fraction first: 2 × 8 + 3 = 19, so 19/8.
Signs and special cases
A negative fraction can carry its sign on the numerator or the denominator; the calculator always moves it to the numerator. A denominator of zero is undefined, and dividing by a fraction whose numerator is zero is impossible, so the result is flagged in both cases.
Converting decimals back to fractions
To turn 0.375 into a fraction, write it over 1,000 and simplify: 375/1000 = 3/8. Repeating decimals such as 0.333… correspond to fractions like 1/3, which have no exact decimal form.