Fractions & Decimals • Topic 3 of 5

Operations with Fractions

The four operations each have their own rule. To add or subtract, you need a common denominator, then combine the numerators (the LCM of the denominators keeps numbers smallest). To multiply, go straight across — numerator × numerator over denominator × denominator — and cancel common factors before multiplying to keep it clean: (3/4)(8/9) becomes (3/4)(8/9) → cancel 3 with 9 and 8 with 4 → (1/1)(2/3) = 2/3. To divide, multiply by the reciprocal: (a/b) ÷ (c/d) = (a/b)(d/c). Mixed numbers should be converted to improper fractions first (2¾ = 11/4) so the rules apply cleanly. The recurring GRE insight: staying in fractions gives exact answers, while the basic calculator forces you into rounded decimals that can cost you a "which is greater" call.

✅ Solved examples

1. Compute 2/3 + 1/4.
Common denominator 12: 2/3 = 8/12, 1/4 = 3/12. Sum = 11/12.
2. Compute (3/4) × (8/9).
Cancel first: 3 and 9 share 3 → 1 and 3; 8 and 4 share 4 → 2 and 1. Left with (1/1)(2/3) = 2/3.
3. Compute (5/6) ÷ (10/3).
Multiply by the reciprocal: (5/6)(3/10) = 15/60 = 1/4.
4. Quantitative Comparison. Quantity A: 1/2 + 1/3 + 1/6. Quantity B: 1. Which is greater?
Common denominator 6: 3/6 + 2/6 + 1/6 = 6/6 = 1. Quantity A = Quantity B; they are equal.

✏️ Practice — try these, take hints as needed

1. Compute 3/5 + 1/2.
Common denominator 10.
6/10 + 5/10.
Add tops.
11/10
2. Compute 7/8 − 1/3.
Common denominator 24.
21/24 − 8/24.
Subtract.
13/24
3. Compute (4/9) × (3/8).
Cancel first.
4 & 8 → 1 & 2; 3 & 9 → 1 & 3.
Multiply.
1/6
4. Compute (2/3) ÷ (4/9).
Multiply by reciprocal.
(2/3)(9/4).
Cancel and simplify.
3/2
5. Compute 2¼ + 1½ as an improper fraction, then simplify.
9/4 + 3/2.
Common denominator 4.
9/4 + 6/4.
15/4 (= 3¾)

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Comparing Fractions Decimals & Place Value →