Fractions are where a lot of people decided they were “bad at maths”, usually around age ten, and the decision stuck. The rules aren’t hard; they were just taught as procedures without the reason behind them, so they never became intuitive. This guide gives the reason for each rule, the four operations with worked examples, the mixed-number and decimal conversions that come up in cooking, construction and finance, and a way to check any answer.
What a fraction is
3⁄4 means three parts, each of which is one-quarter of a whole. The bottom number (denominator) says how big the parts are; the top (numerator) says how many you have. That single idea explains every rule below: you can only count parts together when they are the same size.
Adding and subtracting
Same denominator: add the numerators, keep the denominator. 1⁄8 + 5⁄8 = 6⁄8 = 3⁄4.
Different denominators: convert to a common one first, because you cannot add quarters to thirds directly.
1⁄4 + 1⁄3: the smallest number both 4 and 3 divide into is 12. 1⁄4 = 3⁄12 and 1⁄3 = 4⁄12. So 3⁄12 + 4⁄12 = 7⁄12.
Subtraction is identical: 5⁄6 − 1⁄4 → 10⁄12 − 3⁄12 = 7⁄12.
If finding the least common denominator is slow, multiply the two denominators (4 × 3 = 12 here; for 6 and 4 it gives 24 instead of 12, which still works, you just simplify at the end).
Multiplying
Multiply the tops, multiply the bottoms. 2⁄3 × 3⁄5 = 6⁄15 = 2⁄5. No common denominator needed, because “two-thirds of three-fifths” is a scaling, not a counting. Cancel before multiplying to keep numbers small: the 3 on top and the 3 on the bottom cancel, leaving 2⁄1 × 1⁄5 = 2⁄5 directly.
Dividing
Flip the second fraction and multiply. 3⁄4 ÷ 1⁄2 = 3⁄4 × 2⁄1 = 6⁄4 = 1½. The reason: “how many halves fit into three-quarters?” is the same as “three-quarters times two”, because each whole contains two halves.
Simplifying
Divide top and bottom by their greatest common factor. 18⁄24: both divide by 6, giving 3⁄4. If you can’t spot the largest factor, divide by any common factor repeatedly (÷2 → 9⁄12, ÷3 → 3⁄4). A fraction is fully simplified when the only number dividing both is 1. The Fraction Calculator returns every result simplified and shows the steps, which is the useful part when you’re checking homework or a cut list.
Mixed numbers and improper fractions
2¾ (mixed) and 11⁄4 (improper) are the same value. To convert mixed to improper: whole × denominator + numerator, over the denominator: 2 × 4 + 3 = 11, so 11⁄4. Improper to mixed: divide, the quotient is the whole, the remainder goes on top: 11 ÷ 4 = 2 remainder 3, so 2¾.
Do arithmetic in improper form (it avoids the “borrowing” mess in subtraction) and convert back to mixed for the answer if that’s how people expect to read it. 3½ − 1¾ → 7⁄2 − 7⁄4 → 14⁄4 − 7⁄4 = 7⁄4 = 1¾.
Fractions, decimals and percentages
| Fraction | Decimal | Percent | Where you’ll see it |
|---|---|---|---|
| 1⁄2 | 0.5 | 50% | — |
| 1⁄3 | 0.333… | 33.3% | Recipes scaled by a third |
| 1⁄4 | 0.25 | 25% | Quarter-inch, quarterly |
| 1⁄8 | 0.125 | 12.5% | Eighth-inch (timber, drill bits), stock prices historically |
| 1⁄16 | 0.0625 | 6.25% | Tape measures, socket sets |
| 3⁄8 | 0.375 | 37.5% | Bolt sizes, plywood |
| 5⁄8 | 0.625 | 62.5% | Drywall thickness |
| 2⁄3 | 0.666… | 66.7% | Supermajority votes |
Fraction to decimal: divide top by bottom. Decimal to fraction: write the decimal over its place value and simplify (0.375 = 375⁄1000 = 3⁄8). Repeating decimals (0.333…) cannot be written exactly as finite decimals, which is why fractions are still the right tool for exact work: a third of a 900 mm board is exactly 300 mm as a fraction and 299.999… as a decimal. For the percentage step, the Percentage Calculator handles it.
Real-world fraction problems
- Scaling a recipe that serves 4 to serve 6: multiply every quantity by 6⁄4 = 3⁄2. 2⁄3 cup × 3⁄2 = 1 cup.
- Cutting a 96-inch board into pieces of 14¾ inches: 96 ÷ 14¾ = 96 ÷ 59⁄4 = 96 × 4⁄59 = 6.5, so six pieces with 7½ inches left over.
- Adding tape-measure readings: 5⁄8 + 3⁄16 = 10⁄16 + 3⁄16 = 13⁄16 inch.
- Splitting a bill: three people paying 2⁄5, 2⁄5 and the rest: 1 − 4⁄5 = 1⁄5 for the third person.
Checking an answer
Convert both the question and your answer to decimals and see if they agree. 1⁄4 + 1⁄3 = 7⁄12: 0.25 + 0.333 = 0.583, and 7 ÷ 12 = 0.583. Ten seconds, catches almost every slip.
Frequently asked questions
How do I add fractions with different denominators?
Convert both to a common denominator (the smallest number both divide into, or simply their product), add the numerators, then simplify.
Why do you flip the second fraction when dividing?
Dividing by a fraction asks how many of that fraction fit into the first. Each whole contains (denominator ÷ numerator) of them, which is the flipped fraction, so you multiply by it.
How do I turn 0.375 into a fraction?
Write it over its place value: 375⁄1000, then simplify by 125 to get 3⁄8.
What is 2⁄3 as a percentage?
66.67%, since 2 ÷ 3 = 0.6667. It never ends exactly, so it is usually written as 66.7% or 66⅔%.
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