Fractions & Decimals
1. Core Idea
A fraction is a division waiting to happen. Every fraction question tests one of three skills:
- Operating with fractions (add, subtract, multiply, divide)
- Comparing / ordering fractions
- Converting between fraction, decimal and percent
On the GRE, fractions are almost always faster than decimals. Resist the urge to convert to decimals early — you lose exactness and gain arithmetic.
2. Must-Know Rules
| Operation | Rule |
|---|---|
| a/b + c/d | (ad + bc) / bd |
| a/b - c/d | (ad - bc) / bd |
| a/b x c/d | ac / bd |
| a/b / (c/d) | a/b x d/c (flip and multiply) |
| (a/b)^n | a^n / b^n |
| Compare a/b vs c/d | cross-multiply: ad vs bc (valid when b, d > 0) |
| Mixed to improper | a b/c = (ac + b)/c |
Fraction size intuition:
- Same numerator, bigger denominator -> smaller fraction. 3/7 < 3/5.
- Same denominator, bigger numerator -> bigger fraction. 5/7 > 3/7.
- Adding the same positive k to numerator and denominator of a proper fraction (< 1) moves it closer to 1, i.e. makes it bigger. 2/3 < 3/4 < 4/5.
- For an improper fraction (> 1), adding k to both moves it down toward 1. 5/3 > 6/4 > 7/5.
Terminating vs recurring decimals:
A fraction in lowest terms terminates if and only if its denominator's prime factorisation contains only 2s and/or 5s.
- 7/40 = 7/(2^3 x 5) -> terminates (0.175)
- 5/12 = 5/(2^2 x 3) -> recurring (0.41666...)
Decimal essentials:
- Multiplying decimals: multiply as integers, then the answer has as many decimal places as the two factors combined. 0.03 x 0.4 -> 3 x 4 = 12, three decimal places -> 0.012.
- Dividing decimals: shift both decimal points right until the divisor is a whole number.
- 0.999... = 1 exactly. This is not an approximation.
3. Worked Examples
Example 1 — Complex fraction
Simplify (1/2 + 1/3) / (1/4).
Numerator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Divide: (5/6) / (1/4) = 5/6 x 4/1 = 20/6 = 10/3.
Example 2 — Ordering without decimals
Order 5/8, 7/11, 2/3 from smallest to largest.
Compare 5/8 vs 7/11: cross-multiply -> 5 x 11 = 55 vs 7 x 8 = 56. Since 55 < 56, 5/8 < 7/11.
Compare 7/11 vs 2/3: 7 x 3 = 21 vs 2 x 11 = 22. Since 21 < 22, 7/11 < 2/3.
Order: 5/8 < 7/11 < 2/3.
Example 3 — Terminating test
Which of 3/16, 9/24, 11/30 have terminating decimal forms?
- 3/16 = 3/2^4 -> only 2s -> terminates.
- 9/24 reduces to 3/8 = 3/2^3 -> terminates. (Must reduce first!)
- 11/30 = 11/(2 x 3 x 5) -> has a 3 -> recurs.
3/16 and 9/24 terminate.
4. GRE Traps
- Adding across. 1/2 + 1/3 is not 2/5. You must find a common denominator.
- Forgetting to reduce before testing for terminating decimals. 9/24 looks like it has a 3 in the denominator; reduced, it does not.
- Flipping the wrong fraction when dividing. Flip the second one (the divisor).
- Squaring a fraction between 0 and 1 makes it smaller. (1/2)^2 = 1/4 < 1/2. Students expect powers to grow. Similarly, the square root of a number between 0 and 1 is larger than the number.
- Negative fractions reverse intuition. -1/2 > -3/4, because it is closer to zero.
- Decimal place miscount in multiplication — the classic 0.2 x 0.3 = 0.6 error (it is 0.06).
5. Speed Tricks
- Benchmark against 1/2. To compare fractions quickly, ask whether each is above or below 1/2, 1/3, or 1.
- Cancel before multiplying. (14/25) x (75/28) -> cancel 14/28 = 1/2 and 75/25 = 3 -> 3/2.
- Cross-multiply to compare — it beats converting to decimals every time.
- Keep answers as fractions until the very last step, especially in numeric-entry questions that accept fraction answers.
- Common recurring decimals: 1/3 = 0.333, 1/6 = 0.1666, 1/7 = 0.142857 repeating, 1/9 = 0.111, 1/11 = 0.0909.
6. Self-Check
Q1. Which is larger, 9/13 or 11/16?
Q2. Simplify: (3/4) / (9/8).
Q3. Does 21/56 have a terminating decimal expansion?
Answers
A1. Cross-multiply: 9 x 16 = 144 vs 11 x 13 = 143. Since 144 > 143, 9/13 is larger.
A2. 3/4 x 8/9 = 24/36 = 2/3.
A3. 21/56 reduces to 3/8 (divide both by 7), and 8 = 2^3. Yes — it terminates (0.375).
7. One-Line Summary for the Board
Stay in fractions. Cross-multiply to compare. Reduce before you judge a denominator.