Adding and subtracting fractions
Fractions can only be added when they share a denominator, because the denominator defines the size of the pieces you are counting. To add 1/2 and 1/3, convert both to sixths: 3/6 + 2/6 = 5/6.
The reliable method is cross-multiplication: a/b + c/d = (ad + cb) / bd. This always works, though it may not give the lowest common denominator, which is why the result should then be simplified. Subtraction follows the identical pattern with a minus sign.
| Operation | Working | Result |
|---|---|---|
| 1/2 + 1/3 | Common denominator 6: 3/6 + 2/6 | 5/6 |
| 1/2 − 1/3 | 3/6 − 2/6 | 1/6 |
| 1/2 × 1/3 | 1×1 over 2×3 | 1/6 |
| 1/2 ÷ 1/3 | 1/2 × 3/1 = 3/2 | 3/2 = 1 1/2 |
Multiplying and dividing
Multiplication is the easy case: multiply numerators together and denominators together. 2/3 × 3/4 = 6/12, which simplifies to 1/2. No common denominator is needed.
Division means multiplying by the reciprocal — flip the second fraction and multiply. 1/2 ÷ 3/4 becomes 1/2 × 4/3 = 4/6 = 2/3. This works because dividing by 3/4 asks how many three-quarters fit into the first fraction, which is the same as multiplying by four-thirds.
| To | Do this | Formula |
|---|---|---|
| Add | Same denominator, then add the tops | a/b + c/d = (ad + cb)/bd |
| Subtract | Same denominator, then subtract the tops | a/b − c/d = (ad − cb)/bd |
| Multiply | Tops together, bottoms together | a/b × c/d = ac/bd |
| Divide | Flip the second and multiply | a/b ÷ c/d = ad/bc |
| Simplify | Divide top and bottom by their GCD | 8/12 → 2/3 |
| Mixed → improper | Whole × denominator + numerator | 2 1/3 → 7/3 |
Simplifying and mixed numbers
To reduce a fraction to lowest terms, divide the numerator and denominator by their greatest common divisor. For 6/12 the GCD is 6, giving 1/2. The Euclidean algorithm finds the GCD efficiently: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero.
An improper fraction, where the numerator exceeds the denominator, can be written as a mixed number. Divide to get the whole part and keep the remainder over the original denominator: 7/3 becomes 2 1/3. Improper fractions are usually easier to work with in calculations, while mixed numbers read more naturally in measurements and recipes. The calculator's mixed-number mode converts each to an improper fraction, operates, and converts back.
Converting between fractions and decimals
Dividing the numerator by the denominator gives the decimal. Fractions whose denominators have only 2 and 5 as prime factors terminate — 3/8 is exactly 0.375. Any other denominator produces a repeating decimal: 1/3 is 0.333… and 1/7 is 0.142857 repeating.
Going the other way, a terminating decimal becomes a fraction over a power of ten, then simplifies: 0.75 = 75/100 = 3/4. This is also why exact fraction arithmetic is preferable to decimals when precision matters — 1/3 has no exact decimal representation, so rounding errors accumulate in a way they do not with fractions.
| Decimal | Over a power of ten | Lowest terms |
|---|---|---|
| 0.5 | 5/10 | 1/2 |
| 0.25 | 25/100 | 1/4 |
| 0.75 | 75/100 | 3/4 |
| 0.2 | 2/10 | 1/5 |
| 0.6 | 6/10 | 3/5 |
| 0.125 | 125/1000 | 1/8 |
| 0.375 | 375/1000 | 3/8 |
| 0.0625 | 625/10000 | 1/16 |
| 1.5 | 15/10 | 3/2 |
| 2.75 | 275/100 | 11/4 |
Common fractions, decimals and percentages
Recognising these on sight saves a great deal of arithmetic. The ones with repeating decimals — thirds and sixths — are exactly the ones worth keeping as fractions.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333333… | 33.33…% |
| 2/3 | 0.666667… | 66.67…% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.166667… | 16.67…% |
| 5/6 | 0.833333… | 83.33…% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 1/12 | 0.083333… | 8.333% |
| 1/16 | 0.0625 | 6.25% |