How to Add, Divide, and Simplify Fractions (With Examples)
Common denominators, the reciprocal trick for division, and how to reduce any fraction to lowest terms — with worked examples you can check instantly.
Fractions are one of those topics people learn at ten, forget by twenty, and then meet again in a recipe. Half of 3/4 cup of flour. A 5/8-inch bolt next to a 11/16 wrench. A plank that needs to be 2 1/3 times longer than the offcut. The rules are simple, but there are four of them, and they refuse to stay memorized.
Here’s the whole toolkit, with examples. If you’d rather skip the arithmetic, the Fraction Calculator does all of this as you type — including mixed numbers and decimals.
Adding and subtracting: find a common denominator
You can only add fractions that are cut into the same size pieces. 1/2 + 1/4 makes no sense until you rewrite 1/2 as 2/4. Then it’s easy: 2/4 + 1/4 = 3/4.
The mechanical recipe: multiply each numerator by the other fraction’s denominator, add the results, and put them over the product of both denominators.
- 2/3 + 1/5 → (2×5 + 1×3) / (3×5) = 13/15
- 5/6 − 1/4 → (5×4 − 1×6) / (6×4) = 14/24, which reduces to 7/12
That last step matters. The recipe often produces a bloated fraction, so plan on simplifying afterwards.
Multiplying is the easy one
Multiply straight across: numerators together, denominators together. 2/3 × 3/4 = 6/12 = 1/2. No common denominator needed. If one of the numbers is whole, treat it as a fraction over 1: 4 × 2/3 = 8/3.
Dividing: flip the second fraction
Division is where most people hesitate. The trick is that dividing by a fraction is the same as multiplying by its reciprocal — flip the second fraction upside down and multiply.
1/2 ÷ 3/4 becomes 1/2 × 4/3 = 4/6 = 2/3.
Why does this work? “How many 3/4s fit into 1/2?” is the question being asked. Fewer than one, clearly — and 2/3 of a 3/4 portion is exactly 1/2. The flip isn’t a hack; it’s what division means.
One rule has no workaround: you can’t divide by zero. If the second fraction is 0, there is no answer, and any honest calculator will tell you so instead of inventing one.
Simplifying to lowest terms
A fraction is in lowest terms when the numerator and denominator share no common factor. To get there, divide both by their greatest common divisor (GCD).
Take 84/126. Both divide by 2 → 42/63. Both divide by 21 → 2/3. Done in two steps, though finding that 21 by eye takes practice. The GCD route does it in one: gcd(84, 126) = 42, and 84÷42 over 126÷42 is 2/3 directly.
Teachers insist on lowest terms for a practical reason: 2/3 and 42/63 are the same number, but only one of them is instantly comparable to 3/4.
Mixed numbers and decimals
A mixed number like 2 1/3 is just shorthand for 7/3 — multiply the whole part by the denominator and add the numerator. Convert to improper form before doing any arithmetic, then convert back at the end if you want the readable version.
Decimals go the other way: 0.75 is 75/100, which reduces to 3/4. Terminating decimals always convert exactly. Repeating ones don’t — 1/3 is 0.333333 forever, so any decimal you write down is an approximation.
Try a few of your own in the Fraction Calculator — it accepts fractions, mixed numbers, whole numbers, and decimals, simplifies every answer, and shows the decimal form alongside.