Fractions look simple until the denominators disagree. Ask a room of students for 1/2 + 1/3 and a confident hand will offer 2/5 — add the tops, add the bottoms, done. It is wrong, and the reason it is wrong is the reason fractions are worth understanding properly: a denominator is not a number to count. It is the size of the piece. This post explains adding, simplifying, and mixed numbers with exact numbers, starting with that trap.
What a fraction actually means
A fraction a/b means a parts out of b equal parts. In 3/4, the whole is cut into 4 equal parts and you have 3. The bottom number, the denominator, names the piece size. The top number, the numerator, counts the pieces. Everything in fraction arithmetic follows from that one distinction, and every common mistake comes from forgetting it.
A proper fraction, such as 3/4, is smaller than 1. An improper fraction, such as 7/4, is larger than 1. A mixed number, such as 1 3/4, is the same value as 7/4, because 4/4 + 3/4 = 7/4.
The 1/2 + 1/3 trap
Here is the trap in full. Adding numerators and denominators gives (1 + 1)/(2 + 3) = 2/5. A size check kills it instantly: 1/2 is 0.5 and 1/3 is about 0.333, so the sum must be about 0.833. But 2/5 is 0.4 — smaller than one half alone. A sum of two positive numbers cannot be smaller than one of its parts, so 2/5 is impossible.
The correct method matches piece sizes first. The LCD of 2 and 3 is 6. Convert: 1/2 = 3/6 and 1/3 = 2/6. Add the numerators: 3/6 + 2/6 = 5/6. The GCD of 5 and 6 is 1, so 5/6 is already in lowest terms. Calcrio's free fraction calculator shows this step for any pair of fractions.
The general rule is those four steps: find the least common denominator, convert, add the numerators, and simplify. Our guide on how to add fractions has more examples. Subtraction works the same way: 2/3 − 1/4 = 8/12 − 3/12 = 5/12, because the LCD of 3 and 4 is 12.
Multiplying and dividing: the short versions
Multiplication is the easy operation: multiply straight across, then simplify. For 3/4 × 2/3, multiply numerators (3 × 2 = 6) and denominators (4 × 3 = 12), giving 6/12, which simplifies by GCD 6 to 1/2. You can also cancel before multiplying — the 3s cancel — and reach 1/2 directly. The size check confirms it: 0.75 × 0.667 = 0.5.
Division uses a single memorable move: keep, change, flip. Keep the first fraction, change division to multiplication, and flip the second fraction. For 5/6 ÷ 2/3: 5/6 × 3/2 = 15/12, which simplifies by GCD 3 to 5/4, or 1 1/4 as a mixed number. Only the second fraction flips; flipping the first instead gives 4/5, which is wrong. One more sanity check helps here: dividing by a fraction smaller than 1 gives a larger answer, and 1 1/4 is indeed larger than 5/6.
Simplifying: divide by the GCD
Simplifying means dividing the numerator and denominator by their greatest common divisor, the largest whole number that divides into both. For 12/18, the GCD is 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12/18 = 2/3. The value never changes, because dividing top and bottom by the same number is dividing by 1 in disguise. The fraction is in lowest terms when the numerator and denominator share no common divisor except 1.
More examples: 8/12 simplifies by GCD 4 to 2/3; 15/25 by GCD 5 to 3/5; 9/12 by GCD 3 to 3/4. You can also simplify in stages — 12/18 by 2 to 6/9, then by 3 to 2/3 — as long as you keep going until no common divisor larger than 1 remains.
Mixed numbers: convert first, convert back
A mixed number is easiest to work with as an improper fraction. Convert: 1 1/2 = 3/2 and 2 1/4 = 9/4. Then add: LCD 4, so 3/2 = 6/4, and 6/4 + 9/4 = 15/4. Convert back: 15 ÷ 4 = 3 remainder 3, so 15/4 = 3 3/4. Convert-first prevents the common error of adding whole and fraction parts carelessly.
Converting an improper fraction on its own works the same way: 7/4 becomes 1 3/4, because 7 ÷ 4 = 1 remainder 3. Check it backwards: 1 3/4 = 4/4 + 3/4 = 7/4.
Try it yourself
Pick a problem and work it by hand, then check it. 1/2 + 1/3 = 5/6 by sixths. 3/4 × 2/3 = 1/2 by cancelling. 5/6 ÷ 2/3 = 1 1/4 by keep-change-flip. 12/18 = 2/3 by GCD 6. 1 1/2 + 2 1/4 = 3 3/4 by converting first. And if you want the full steps shown — the common denominator, the conversion, the simplification — the free fraction calculator on Calcrio does exactly that, with no sign-up.
Fractions reward one habit: check the size. If an answer is wildly off your decimal estimate, the error is almost always in the denominator step.
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