Add, Subtract, Multiply & Divide Fractions

The order that never changes

Fractions do not get their own order of operations. A product behaves like a single amount — three groups of four is one number, 12 — so every multiplication in the problem resolves before any addition touches it. The same logic that makes 3 + 4 × 5 equal 23 makes 1/2 + 3/4 × 2/3 equal 1.

Work the wrong way around and the answer drifts: adding first gives 5/4, and 5/4 × 2/3 lands on 5/6. Nothing about either fraction is difficult — the sequence was. When a mixed problem comes out wrong, the first thing to check is not the arithmetic but the order it ran in.

Division inside a longer expression

A division step follows the same keep-change-flip rule it always does, whenever its turn arrives. In the problem below, the division resolves first and leaves a cleaner subtraction behind it:

Notice the middle line: 3/2 − 1/4 has unlike denominators, so the subtraction pauses for a common-denominator rewrite before finishing. Mixed problems alternate between the two skills this way — the order of operations tells you which skill is next, and each skill runs exactly as it does on its own page.

Simplify as you go

An unsimplified middle result is not a mistake, but it is extra weight. Carrying 6/12 into an addition step means rewriting it to match denominators later anyway. Simplifying 6/12 to 1/2 immediately keeps every following denominator small — and small numbers are where fraction arithmetic is easiest to get right.

There is one hard rule at the end: the final answer must be in lowest terms, whatever happened in the middle. If the last line reads 5/4, that is finished; if it reads 10/8, one more division by 2 is part of the problem.

Mixed numbers and parentheses

Two situations adjust the routine without changing it. A mixed number converts to an improper fraction before its first operation — 1 1/2 becomes 3/2 — because a whole part and a fractional part cannot multiply or divide in mixed form. And parentheses always outrank everything: whatever sits inside them is one operation, finished before the expression around it continues.

Common mistakes in mixed fraction problems

• Adding before multiplying: In 1/2 + 3/4 × 2/3, adding first gives 5/4 × 2/3 = 5/6 — the wrong path. Multiplication runs first: 3/4 × 2/3 = 1/2, then 1/2 + 1/2 = 1. One misplaced step changes the entire answer.
• Finding common denominators for multiplication or division: Matching denominators belongs to addition and subtraction only. In 2/3 × 3/5, matching the denominators first wastes effort and invites mistakes; multiplying across reaches 2/5 directly.
• Flipping the first fraction in a division step: Inside a longer expression the division rule does not change: keep the amount you have, flip only the divisor. In 3/4 ÷ 1/2 × 1/3, the 3/4 stays put while 1/2 becomes 2/1.
• Skipping simplification between steps: An unsimplified middle result is not wrong, but it makes everything after it harder. 3/4 × 2/3 leaves 6/12; carrying 6/12 into the next step means adding 6/12 + something. Simplify to 1/2 first and the next step shrinks.