How Fraction Arithmetic Works
A fraction represents a part of a whole. The top number (numerator) tells you how many parts you have, and the bottom number (denominator) tells you how many equal parts make up the whole. Fraction arithmetic follows four distinct sets of rules depending on the operation.
For addition and subtraction, both fractions must share the same denominator before you can combine them. You find the least common denominator, adjust both numerators accordingly, then add or subtract the numerators while keeping the denominator unchanged.
For multiplication, you multiply numerators together and denominators together. No common denominator is needed. For division, you flip the second fraction (find its reciprocal) and then multiply.
Leah at her Pinewood Falls bakery uses fraction arithmetic constantly. When she scales a recipe that calls for 2/3 cup of sugar to make 1½ batches, she calculates 2/3 × 3/2 = 6/6 = 1 cup. She uses the fraction to decimal converter when she needs to set her digital scale.
The Fraction Formulas
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×d) / (b×c)
These cross-multiplication formulas always work, but using the LCD (least common denominator) for addition and subtraction produces smaller numbers that are easier to simplify.
Maya is studying fractions for a placement test. She needs to add 3/8 + 5/12. Using the LCD method: LCD of 8 and 12 is 24. Convert: 9/24 + 10/24 = 19/24. Using cross multiplication: (3×12 + 5×8) / (8×12) = (36 + 40) / 96 = 76/96 = 19/24. Both methods reach the same answer, but the LCD approach avoids simplifying at the end.
Finding the Least Common Denominator
The least common denominator (LCD) is the smallest number that both denominators divide into evenly. It equals the least common multiple (LCM) of the two denominators. There are two common methods to find it:
| Method | Steps | Example (denominators 8 and 12) |
|---|---|---|
| List multiples | List multiples of each denominator until you find the first common one | 8: 8, 16, 24... / 12: 12, 24... → LCD = 24 |
| GCD method | LCD = (a × b) / GCD(a, b) | GCD(8, 12) = 4. LCD = (8 × 12) / 4 = 24 |
The GCD method is faster for large numbers. The GCD (greatest common divisor) can be found using the Euclidean algorithm.
After finding the LCD, multiply each fraction's numerator by the factor needed to bring its denominator up to the LCD. For 3/8 with LCD 24: multiply numerator and denominator by 3 (since 24 ÷ 8 = 3), giving 9/24. For 5/12: multiply by 2 (since 24 ÷ 12 = 2), giving 10/24.
Simplifying Fractions
A fraction is in simplest form when the numerator and denominator have no common factors other than 1. To simplify, divide both by their greatest common divisor (GCD). The Euclidean algorithm finds the GCD efficiently: repeatedly divide the larger number by the smaller and take the remainder, until the remainder is 0. The last non-zero remainder is the GCD.
| Original Fraction | GCD | Simplified | Decimal |
|---|---|---|---|
| 4/8 | 4 | 1/2 | 0.5 |
| 6/9 | 3 | 2/3 | 0.6667 |
| 12/18 | 6 | 2/3 | 0.6667 |
| 15/25 | 5 | 3/5 | 0.6 |
| 24/36 | 12 | 2/3 | 0.6667 |
| 35/49 | 7 | 5/7 | 0.7143 |
Decimals rounded to 4 places. Some fractions (like 1/3, 1/7) produce repeating decimals that cannot be expressed exactly.
Common Fraction–Decimal Equivalents
The table below shows commonly used fractions and their decimal and percentage equivalents. These come up frequently in cooking, construction, and everyday math.
| Fraction | Decimal | Percent | Fraction | Decimal | Percent |
|---|---|---|---|---|---|
| 1/2 | 0.5 | 50% | 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33.3% | 3/8 | 0.375 | 37.5% |
| 2/3 | 0.667... | 66.7% | 5/8 | 0.625 | 62.5% |
| 1/4 | 0.25 | 25% | 7/8 | 0.875 | 87.5% |
| 3/4 | 0.75 | 75% | 1/16 | 0.0625 | 6.25% |
| 1/5 | 0.2 | 20% | 1/10 | 0.1 | 10% |
Fractions with denominators that are powers of 2 or 5 produce terminating decimals. All others produce repeating decimals.
Dana frequently uses fractions in her Pinewood Falls construction projects. Lumber is measured in fractions of an inch. A 2×4 is actually 1 1/2 × 3 1/2 inches. When she needs to cut a board into thirds and account for 1/8-inch saw blade kerf, she subtracts 2/8 (two cuts) from the total length, then divides the remainder by 3. She uses this calculator for the arithmetic and the decimal to fraction converter when reading measurements from her digital caliper. The percentage calculator is also useful when converting fractions to percentages for estimates and quotes.
This calculator performs exact fraction arithmetic using integer numerators and denominators. Results are mathematically precise, but very large numerators or denominators may exceed JavaScript's safe integer range (253 − 1). For everyday fractions this is not a concern.