Fraction addition

Divide the numerator and denominator by their greatest common divisor (GCD) — the largest number that goes into both evenly — and you're left with the same fraction in its smallest possible form.

1. Start with 3/4.
2. GCD of 3 and 4 is 1.
3. Divide both by it: 3/4 = 3/4.

If the GCD comes out to 1, there's nothing left to cancel — the fraction was already in lowest terms.

Addition and subtraction only work once both fractions describe the same size of piece — you can't add “3 quarters” and “5 sixths” directly any more than you can add 3 apples and 5 oranges and call the answer 8 of either. The cleanest common denominator is the least common multiple (LCM) of the two denominators. Any common multiple technically works — multiplying the two denominators together (4 × 6 = 24) always gives you one — but the LCM keeps the numbers smaller and means less simplifying once you're done.

Once both fractions share a denominator, just add or subtract the numerators:

• Addition: 3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 19/12.
• Subtraction: 3/4 − 5/6 = 9/12 − 10/12 = -1/12, which simplifies to 1/12 (negative — 5/6 is the larger fraction here).

Neither of these needs a common denominator, which makes them the quicker pair of the four.

• Multiplication: numerators together, denominators together — 3/4 × 5/6 = 15/24 = 5/8.
• Division: multiply by the reciprocal of the second fraction — 3/4 ÷ 5/6 = 3/4 × 6/5 = 18/20 = 9/10.

The shorthand for division is “keep, change, flip” — keep the first fraction as is, change ÷ to ×, flip the second fraction. It works because dividing by a number and multiplying by its reciprocal are the same operation, the same way dividing by 5 and multiplying by 1/5 land on the same answer. If you're dividing by a whole number instead of a fraction, the same trick still applies — a whole number like 5 is really just 5/1, and its reciprocal is 1/5.

Fraction to decimal: divide numerator by denominator. Decimal to percentage: multiply by 100. Going backwards from a percentage, write it over 100 and simplify — 75% = 75/100 = 3/4.

Whether a decimal ends cleanly or repeats forever comes down to the denominator once the fraction is simplified. 3/4 reduces to 3/4, and because 4 breaks down into only 2s and 5s, its decimal terminates cleanly — exactly what 0.7500 shows above. 5/6 simplifies to 5/6, whose decimal repeats too, for the same reason. Any denominator built purely from 2s and 5s divides evenly into some power of 10; anything else — a 3, a 7, an 11 — never does, no matter how many digits you write.

Decimals are also the fastest way to compare two fractions with different denominators without doing common-denominator arithmetic by hand: 3/4 = 0.7500 against 5/6 = 0.8333 means 5/6 is the larger of the two.

Converting to decimals is fastest when you have one handy, but three other tricks work just as well by hand, and they're what most textbooks actually teach:

• Cross-multiply. For a/b versus c/d, compare a×d against b×c — whichever product is bigger tells you which fraction is bigger. It's a shortcut for putting both fractions over the same denominator without actually writing that denominator out.
• Common denominator. Convert both fractions the way you would for addition, then just compare the numerators directly.
• Benchmark against 1/2. If one fraction is clearly above 1/2 and the other clearly below it, you already have your answer with no arithmetic at all — useful for a quick sanity check before you commit to the longer method.

A mixed number like 2 3/4 pairs a whole number with a proper fraction. An improper fraction, like 11/4, has a numerator bigger than its denominator. They're the same value, two different ways of writing it:

• Mixed → improper: multiply the whole number by the denominator, add the numerator, keep the denominator — 2 3/4 = (2×4+3)/4 = 11/4.
• Improper → mixed: divide numerator by denominator — the quotient becomes the whole number, the remainder sits over the original denominator — 11/4 = 2 remainder 3 = 2 3/4.

Improper fractions are easier to calculate with; mixed numbers are easier to read afterward — “2 3/4 cups” means more to a cook than “11/4 cups” does. Convert to improper before doing the arithmetic, then convert back at the end if the mixed form is what you actually need.

These three show up together constantly in school problems, and mixing them up is where a lot of word problems go sideways. A fraction is a part of one whole — 3 out of 4 equal pieces. A ratio compares two separate quantities that don't have to come from the same whole, like 3 cups of flour to 2 cups of sugar. A proportion is a statement that two ratios are equal, and it's what you actually solve when a recipe question asks how much flour you'd need to keep the same taste at a different batch size. All three lean on the same arithmetic covered above — once you can add, simplify, and cross-multiply fractions confidently, ratio and proportion problems stop feeling like a separate topic.

Say a recipe uses 3/4 cup of an ingredient and serves 4, but you're cooking for 6. The scale factor is 6/4, which simplifies to 3/2 — multiply the original quantity by that.

1. Original quantity: 3/4 cup.
2. Scale factor: 6 servings ÷ 4 servings = 6/4 = 3/2.
3. New quantity: 3/4 × 3/2 = 9/8, simplified to 9/8 cup.

Same move works for resizing a blueprint or splitting a shared cost unevenly — multiply the known quantity by whatever fraction describes the new proportion.

Division comes up less often than multiplication in daily life, but it's the natural operation whenever you're asking “how many of these fit into that.” Say you have 3/4 of something — a length of ribbon, a tank of fuel, a block of time — and you want to know how many 5/6-sized portions it splits into.

1. Set it up as division: 3/4 ÷ 5/6.
2. Flip and multiply: 3/4 × 6/5 = 18/20.
3. Simplify: 18/20 = 9/10, so you get 9/10 full 5/6-sized portions out of the original 3/4.

Most fraction errors trace back to one of a handful of habits, and they're worth checking for specifically once you have an answer:

• Adding numerators and denominators separately without finding a common denominator first — this is the single most common mistake in school, and it produces an answer that looks plausible enough to slip past a quick check.
• Forgetting to simplify the final answer. The arithmetic can be entirely correct and still lose marks if 15/24-style output is left un-reduced instead of written as 5/8.
• Flipping the wrong fraction when dividing — only the second fraction, the divisor, ever gets flipped. Flip the first one instead and the answer is the reciprocal of what it should be.
• Treating a mixed number like 2 1/2 as 2/1/2 instead of converting it to an improper fraction before calculating — mixed numbers can't be multiplied or divided directly as written.
• Assuming a bigger denominator means a smaller fraction. That's only true when the numerators match. 5/6 is bigger than 1/2 despite the larger denominator, because the numerator matters just as much.
• Rounding a repeating decimal too early in a multi-step calculation — the error compounds through every step that follows. Keep the fraction form as long as you can, and round only at the very end.

The usual instinct when a child gets stuck is to re-explain the algorithm — find the common denominator, cross-multiply, and so on — a bit more slowly. That often doesn't help, because the child never had a concrete picture of what a fraction is in the first place. Cutting an actual roti or a paper circle into equal wedges, or using measuring cups while cooking, gives the idea somewhere to live before the symbols show up. Once “equal-sized pieces of a whole” feels obvious, the rule about needing a common denominator stops being an arbitrary thing to memorise and starts being the only way the arithmetic could possibly work — which is also what stops a child from adding numerators and denominators separately, the mistake almost every learner makes at least once.

Worth noticing: the sum and the product don't need to come out looking anything alike, even though they're built from the same two starting fractions — addition and multiplication measure genuinely different things, so there's no shortcut from one result to the other.

Figures above are computed live from the fractions you entered (or the illustrative defaults shown), using the Euclidean algorithm for GCD and the standard LCM-based method for common denominators — the same methods most school curricula teach, so the steps here should line up with a textbook regardless of board or grade level. Decimal values are shown to four places and percentages to one; the terminating- versus-repeating check looks only at the prime factors of the simplified denominator, which is the standard test and holds for any fraction, not just the two used as the example here.