Fractions without the guesswork
A fraction is a division waiting to happen: 3/4 means 3 ÷ 4. Everything else follows from that. Adding and subtracting need the pieces to be the same size, so you rewrite both fractions over a common denominator, combine the numerators, and simplify. Multiplying and dividing skip that step entirely — multiply straight across, or flip the second fraction and multiply.
Adding and subtracting: find the least common denominator
Take 1/3 + 1/6. The denominators 3 and 6 have a least common multiple of 6, so 1/3 becomes 2/6 (multiply top and bottom by 2 — the value never changes). Then 2/6 + 1/6 = 3/6, and since the greatest common divisor of 3 and 6 is 3, that simplifies to 1/2. The classic mistake is adding denominators: 1/3 + 1/6 is not 2/9. Picture a cake — a third plus a sixth is half of it, not a ninth.
Multiplying and dividing
- Multiply: numerator × numerator, denominator × denominator, then simplify. 2/3 × 3/5 = 6/15 = 2/5.
- Divide: multiply by the reciprocal (flip the second fraction). 1/3 ÷ 1/6 = 1/3 × 6/1 = 2. The reason is worth knowing: dividing asks "how many of the second fraction fit into the first", and sixths fit into a third exactly twice.
- Simplify first to keep numbers small: in 4/9 × 3/8 you can cancel the 4 with the 8 and the 3 with the 9 before multiplying, which turns a 12/72 into 1/6 without the arithmetic.
Mixed numbers, improper fractions and decimals
A mixed number such as 2 3/4 is just 11/4 (2 × 4 + 3 over 4). Convert to an improper fraction before multiplying or dividing, and convert back at the end with a division and a remainder. Every fraction is also a decimal: divide the numerator by the denominator. Some divisions terminate (1/4 = 0.25, 3/8 = 0.375), and others repeat forever, which is what the brackets in the output mean: 1/3 = 0.(3), 1/6 = 0.1(6), 1/7 = 0.(142857). Percentages are the same number over 100, so 1/3 = 33.33%. Comparing two fractions never needs decimals — cross-multiply and compare the products: 2/3 against 3/5 gives 10 against 9, so 2/3 is larger.
Where fractions show up in daily life
Recipes are the obvious one: 1/2 cup plus 1/3 cup is 5/6 cup, doubling a 3/4 teaspoon gives 1 1/2 teaspoons, and halving an odd number of thirds forces you to think in sixths. Imperial tape measures are fractions all the way: the marks are halves, quarters, eighths and sixteenths, and adding two lengths means adding fractions with different denominators. Discounts and odds are fractions too — 1/3 off is not 33% off (it is 33.3% off), and 3-to-1 odds mean three chances against one. And in schoolwork, showing the common-denominator step is worth more marks than the answer, which is why this calculator prints the working rather than just the result.
Frequently asked questions
How do I add fractions with different denominators?
Rewrite both over the least common multiple of the denominators, then add the numerators and simplify. 1/3 + 1/6 becomes 2/6 + 1/6 = 3/6 = 1/2. Never add the denominators themselves: 1/3 + 1/6 is 1/2, not 2/9.
How do I divide fractions?
Flip the second fraction (its reciprocal) and multiply: 1/2 ÷ 1/4 = 1/2 × 4/1 = 2. The intuition is that dividing asks how many of the second fraction fit into the first, and two quarters fit into a half.
Why does 1/3 + 1/6 equal 1/2?
Because a third is two sixths, so 1/3 + 1/6 is 2/6 + 1/6 = 3/6, and 3/6 reduces to 1/2 by dividing both parts by their greatest common divisor, 3.
How do I simplify a fraction?
Find the greatest common divisor of the numerator and denominator and divide both by it. For 18/24 the GCD is 6, giving 3/4. If the GCD is 1 the fraction is already in its simplest form.
How do I convert a fraction to a decimal and a percentage?
Divide the numerator by the denominator for the decimal (3/8 = 0.375), then multiply by 100 for the percentage (37.5%). Some fractions never divide exactly and repeat instead: 1/3 = 0.(3) = 33.33%, and the bracket marks the repeating cycle.
How do I turn a decimal into a fraction?
Count the decimal places, put the digits over that power of ten, and simplify. 0.375 = 375/1000 = 3/8. For a mixed value like 2.5, split it into 2 + 0.5 = 2 1/2.
How do I compare two fractions without a calculator?
Cross-multiply: compare numerator × other denominator for both. For 2/3 and 3/5, 2 × 5 = 10 and 3 × 3 = 9, so 2/3 is larger. Alternatively convert both to decimals, but cross-multiplication stays exact.
What is a mixed number and when should I convert it?
A mixed number such as 2 3/4 combines a whole number and a proper fraction, and equals the improper fraction 11/4. Convert to an improper fraction before multiplying or dividing, and convert back afterwards.