FDP (Fractions, Decimals, Percentages) is one of the highest-yield areas on the GMAT. Mastery lets you move quickly and avoid arithmetic errors across a wide range of question types.

A) Fraction Fundamentals

In a fraction n/d, n is the numerator and d is the denominator. The denominator can never be 0.

• Equivalent fractions: divide numerator and denominator by their GCD to reduce to lowest terms. 8/36 and 14/63 both reduce to 2/9.
• Addition/Subtraction: requires a common denominator. Convert each fraction to an equivalent with the LCM as the denominator, then add/subtract numerators.
• Multiplication: multiply numerators together and denominators together. Cancel common factors before multiplying to simplify. (3/8) × (4/9) = 12/72 = 1/6.
• Division: multiply by the reciprocal. (a/b) ÷ (c/d) = (a/b) × (d/c).
• Comparing fractions: cross-multiply (keep track of direction). 3/7 vs. 4/9 → 27 vs. 28 → 4/9 is larger.

B) Mixed Numbers

A mixed number like 3½ means 3 + ½. Convert to an improper fraction before doing arithmetic: multiply the whole number by the denominator, add the numerator, keep the denominator. 3½ = (3×2 + 1)/2 = 7/2. To convert back: divide numerator by denominator, the quotient is the whole part, remainder is the new numerator.

C) Decimals

• Multiplying: multiply as integers, then insert decimal point so the total digits after the decimal = sum of decimal places in both factors. 2.09 × 1.3 = 209 × 13 = 2717, then 3 decimal places → 2.717.
• Dividing: move decimal points equally in dividend and divisor to make the divisor an integer, then divide normally. 698.12 ÷ 12.4 = 6981.2 ÷ 124.
• Scientific notation: 231 = 2.31 × 10². Moving decimal right (positive exponent), left (negative exponent).

D) Terminating vs. Repeating Decimals

A fraction a/b in lowest terms produces a terminating decimal if and only if the denominator has no prime factors other than 2 and 5 — i.e., b = 2^m × 5ⁿ for some m, n ≥ 0. Otherwise the decimal repeats.

• 1/4 = 1/(2²) → terminates (0.25). 1/6 = 1/(2×3) → repeating (0.1666…).
• To convert a repeating decimal to a fraction: let x = 0.abcabc…, multiply by 10^k (where k = length of repeating block), then subtract x and solve. Example: x = 0.363636… → 100x = 36.3636… → 99x = 36 → x = 36/99 = 4/11.

E) Key Fraction-Decimal Conversions

• 1/2 = 0.5    1/3 ≈ 0.333    2/3 ≈ 0.667    1/4 = 0.25    3/4 = 0.75
• 1/5 = 0.2    1/6 ≈ 0.167    1/7 ≈ 0.143    1/8 = 0.125    3/8 = 0.375    5/8 = 0.625
• 1/9 ≈ 0.111    1/10 = 0.1    1/11 ≈ 0.091    1/12 ≈ 0.083

F) Percentages

"Percent" means "per hundred." Core formula: x% of y = (x/100) × y.

• Percent increase: (new − old) / old × 100%
• Percent decrease: (old − new) / old × 100%
• Successive changes: +20% then −20% ≠ 0. Result: 1.2 × 0.8 = 0.96 → net 4% decrease. Multiply the multipliers.
• Percentage points vs. percent change: going from 20% to 25% is +5 percentage points but a 25% increase in the percent itself (5/20 = 0.25).
• "A is 30% more than B" → A = 1.3B. "A is 30% less than B" → A = 0.7B.

G) Interest Calculations

Interest is the extra money earned (or owed) on a principal amount over time.

• Simple interest: I = P × r × t. Interest only on the original principal, never on accumulated interest. Example: $1,000 at 6% for 3 years → I = 1000 × 0.06 × 3 = $180. Total = $1,180.
• Compound interest (annually): A = P(1 + r)^t. Interest is added to the principal each year, then the next year's interest is calculated on the new (larger) total. Example: $1,000 at 6% for 3 years → A = 1000 × (1.06)³ = 1000 × 1.191 = $1,191. Slightly more than simple interest.
• Compound interest (n times per year): A = P(1 + r/n)^nt. If compounded quarterly (n=4) at 6% for 2 years: A = 1000 × (1 + 0.06/4)⁸ = 1000 × (1.015)⁸ ≈ $1,126.49.

H) Ratios and Proportions

• A ratio a:b means for every a units of one, there are b units of the other. Total parts = a + b.
• A proportion is an equation between two ratios: a/b = c/d → cross-multiply: ad = bc.
• Example: if 5 shirts cost $44, then 8 shirts cost c where 5/44 = 8/c → c = (8 × 44)/5 = $70.40.
• Scaling: if a:b = 2:3 and total = 50, then a = (2/5) × 50 = 20, b = 30.

I) Rounding Rules

The GMAT Official Guide's reference section covers rounding rules and estimation techniques used across all question types. Mastering these prevents arithmetic errors and speeds up calculations on questions where an exact answer is unnecessary.

To round a number to the nearest 10^−k (e.g., nearest tenth = 10⁻¹, nearest hundredth = 10⁻²):

1. Identify the target digit: the digit at position 10^−k.
2. Look at the digit immediately to the right of the target digit (one position further right).
3. If that right-neighbor digit is 5 or greater: round the target digit up by 1.
4. If that right-neighbor digit is less than 5: keep the target digit as is (round down = truncate).
5. Drop all digits to the right of the target position.

To round to the nearest integer, look at the tenths digit. If it is ≥ 5, round up. If it is < 5, round down. Example: 7651.4 → 7651; 3.78 → 4.

Multi-step calculations: do not round intermediate results. Rounding too early introduces cumulative error. Round only the final answer, or round strategically to make arithmetic manageable.

J) Estimation Techniques

For multiple-choice questions, the answer choices are often spread far enough apart that an estimate eliminates all but one option.

• Round to 1 significant figure: replace each number with the nearest power of 10 or half-power. 7.8 × 31.2 ≈ 8 × 30 = 240.
• Simplify complex fractions by rounding: replace numerator and denominator with nearby round numbers. (4.97 × 19.2) / (2.03 × 9.8) ≈ (5 × 20) / (2 × 10) = 100/20 = 5. Track whether you rounded both up (overestimate), both down (underestimate), or in opposite directions (partial cancel).
• Estimate by nearest perfect square or cube: √83 is between √81=9 and √100=10, closer to 9. ∛26 is between ∛27=3 and ∛8=2, very close to 3.
• Upper and lower bounds: for expressions inside square roots or fractions, bracket the answer. If √24.82: √25 = 5 and √24 ≈ 4.9, so the answer is between 4.9 and 5.0.
• Percent estimation: 23% of 497 ≈ 25% of 500 = 125. Then adjust downward slightly since both 23% < 25% and 497 < 500.
• Track rounding direction: when both numerator and denominator are rounded in the same direction, errors partially cancel. When rounded in opposite directions, errors compound — always track which direction you rounded each value.

K) Rounding on DS Questions

On Data Sufficiency questions, rounding can affect whether a statement is sufficient:

• If the question asks "what is x rounded to the nearest integer?" and a statement gives 2.3 ≤ x ≤ 2.7, the answer is NOT sufficient — x could round to 2 or 3.
• But if the statement gives 2.6 ≤ x ≤ 2.9, then rounded to the nearest integer x = 3 — sufficient.
• Key: a range that straddles a midpoint (like 2.5) creates ambiguity in rounding → insufficient.

• Rounding in the wrong direction: "round down" means toward zero for positives but away from zero for negatives. −2.7 rounded to nearest integer is −3, not −2.
• Successive rounding: rounding a number twice can give a different result than rounding once. Always round only once to the final target precision.
• Percent vs. percentage-point rounding: 33.3% rounded to the nearest percent is 33%, not 33.3% or 34%.