Mastering Fraction Operations
Adding, subtracting, multiplying, and dividing fractions — the four arithmetic moves, adapted for fractions. Two big ideas run the whole show: for adding and subtracting, the denominators must match; for multiplying, just multiply straight across; for dividing, flip the second fraction and multiply.
1 The Two Big Ideas
Every fraction operation is one of these two patterns. Learn the patterns and the mechanics take care of themselves.
Idea 1 — Adding and subtracting need matching denominators
You cannot add thirds to quarters any more than you can add apples to oranges. So you rename both fractions with equivalent forms that share a denominator (usually the least common multiple), then add or subtract only the tops.
Idea 2 — Multiplying goes straight across; dividing flips
For multiplication, multiply the tops and multiply the bottoms — no common denominator needed. For division, keep the first fraction, change ÷ to ×, and flip the second fraction (its reciprocal).
Why it matters
This is the last “mechanical” fraction skill before fractions dissolve into decimals, percents, ratios, and algebra. Get fluent here and every later topic that touches fractions feels easy.
Where it is used
Doubling a recipe (multiply) · splitting a bill three ways (divide) · combining 1/2 tank + 1/4 tank of gas (add) · woodworking measurements (subtract).
2 Worked Examples
Follow each step. For +/−: common denominator first. For ×: straight across. For ÷: keep, change, flip.
- Thinking: the pieces are already the same size (sevenths), so I add the counts: 2 + 3 = 5 sevenths.
- 2/7 + 3/7 = 57.
- Check: 5/7 is in lowest terms (gcd(5,7) = 1), and the answer is between 0 and 1 — sensible for two small fractions.
- Thinking: quarters and thirds are different piece sizes, so I rename both with denominator 12 (the least common multiple of 4 and 3).
- 1/4 = 3/12 and 2/3 = 8/12, so 3/12 + 8/12 = 1112.
- Check: estimate — 1/4 + 2/3 ≈ 0.25 + 0.67 = 0.92, and 11/12 ≈ 0.917. Matches.
- Thinking: rename 1/3 as 2/6, then subtract: 5/6 − 2/6 = 3/6. Always simplify: 3/6 = 1/2.
- 5/6 − 1/3 = 12.
- Check: 5/6 ≈ 0.833, 1/3 ≈ 0.333, difference ≈ 0.5.
- Thinking: multiply straight across: (2×3)/(3×5) = 6/15. But the 3s cancel first — smarter: 2/3 × 3/5 = 2/5 directly.
- 2/3 × 3/5 = 25.
- Check: 6/15 simplifies by 3 to 2/5. Estimate: (2/3)(3/5) = 0.4.
- Thinking: keep 3/4, change ÷ to ×, flip 2/5 to 5/2. Then 3/4 × 5/2 = 15/8 = 1 7/8.
- 3/4 ÷ 2/5 = 158 = 1 7/8.
- Check: 15/8 = 1.875; 3/4 = 0.75 and 2/5 = 0.4, and 0.75 ÷ 0.4 = 1.875.
- Thinking: convert to improper fractions: 3/2 + 9/4. Common denominator 4: 6/4 + 9/4 = 15/4 = 3 3/4.
- 1 1/2 + 2 1/4 = 154 = 3 3/4.
- Check: 1.5 + 2.25 = 3.75.
3 Common Mistakes
Three errors that show up on almost every fraction quiz. Spot them now and they will never cost you points.
But 2/5 = 0.4 is SMALLER than 1/2 = 0.5 — adding made it shrink!
The denominator names the piece size — rename first, then add only the tops.
Sanity check fails: “how many 0.4s fit in 0.75?” — nearly two, not 0.3!
Division by a fraction smaller than 1 always gives a bigger answer — use that as your check.
Cancel common factors before or after computing, every single time.
4 Quick Check
Try each one on paper first, then reveal the answer.
Key Points to Remember
- Add / subtract: denominators must match — rename with equivalent fractions first, then combine only the numerators.
- Multiply: straight across — (a×c)/(b×d). Cancel common factors before multiplying when you can.
- Divide: keep the first, change ÷ to ×, flip the second. Dividing by a fraction smaller than 1 gives a bigger answer.
- Simplify last — every time, no exceptions.
- Mixed numbers: convert to improper fractions first, then operate.