Arithmetic / FDP

What This Module Covers

This module covers the foundational number skills that appear in nearly every GRE Quant section: converting between fractions, decimals, and percents; calculating percent changes and successive changes; working with ratios, rates, exponents, and absolute values. These are not advanced topics — but the GRE exploits exactly the small conceptual gaps most test-takers have about them.

Why It Matters on the GRE

FDP questions account for a significant share of all GRE Quant problems, and the traps are predictable. Master these mechanics once and you stop losing points to arithmetic you've known since middle school.

Core Concepts

1. Converting Between Fractions, Decimals, and Percents

The GRE moves freely between these three forms, and so should you. To convert a fraction to a decimal, divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75. To convert a decimal to a percent, multiply by 100: 0.75 × 100 = 75%. To go from percent back to a decimal, divide by 100: 35% = 0.35.

Memorizing key benchmarks saves you 10–15 seconds per problem — which adds up fast.

Must Know: Benchmark conversions to memorize cold: 1/8 = 12.5% | 1/6 ≈ 16.7% | 1/4 = 25% | 1/3 ≈ 33.3% | 3/8 = 37.5% | 1/2 = 50% | 2/3 ≈ 66.7% | 3/4 = 75% | 7/8 = 87.5%

Example — Convert 5/8 to a percent:

1. Divide: 5 ÷ 8 = 0.625
2. Multiply by 100: 0.625 × 100 = 62.5%
3. Or use benchmarks: 5/8 = 4/8 + 1/8 = 50% + 12.5% = 62.5% ✓

2. The Percent Formula

Every percent problem is a version of one equation: Part = Percent × Whole. The hard part is identifying which number is the "whole." The whole is always the number that comes right after the word "of" in the question.

Must Know: Part = Percent × Whole. Whole = the number after "of." If you're solving for the percent, rearrange: Percent = Part / Whole.

Example — "What percent of 80 is 20?"

1. Identify: Whole = 80 (comes after "of"), Part = 20
2. Apply: Percent = Part / Whole = 20 / 80 = 0.25
3. Convert: 0.25 × 100 = 25%
4. Verify: 25% × 80 = 0.25 × 80 = 20 ✓

3. Percent Change

Percent change measures how much something grew or shrank relative to where it started. The formula is: Percent Change = (New − Old) / Old × 100. The critical word is "Old" — the original value is always the base (denominator), not the new value.

Must Know: The base in a percent change calculation is ALWAYS the original (old) value. Using the new value as the base is the single most common percent error on the GRE.

Example — A salary rises from $50,000 to $60,000. What is the percent increase?

1. New = 60,000 | Old = 50,000
2. Change = 60,000 − 50,000 = 10,000
3. Percent change = 10,000 / 50,000 × 100 = 20%
4. Verify: 20% of 50,000 = 10,000; 50,000 + 10,000 = 60,000 ✓

4. Successive Percent Changes

This is one of the GRE's favorite traps. When a value goes up by 20% and then down by 20%, most people assume you're back where you started. You're not. Each percent change operates on a different base — the second change uses the already-modified value, not the original.

The correct method: convert each percent change to a multiplier and multiply them together.

Must Know: Successive percent changes MULTIPLY — they do not add. A 20% increase then a 20% decrease = ×1.20 × ×0.80 = ×0.96, which is a net 4% decrease, not 0%.

Example — A 20% increase followed by a 20% decrease:

1. Increase multiplier: 100% + 20% = 1.20
2. Decrease multiplier: 100% − 20% = 0.80
3. Combined: 1.20 × 0.80 = 0.96
4. Net effect: 0.96 − 1.00 = −0.04 = 4% decrease
5. Verify with $100: $100 → ×1.20 → $120 → ×0.80 → $96. Net loss = $4 on $100 = 4% ✓

5. Ratio and the Ratio Multiplier

A ratio like a:b = 3:5 does not mean a = 3 and b = 5. It means a and b are in proportion 3 to 5. The actual values are 3k and 5k for some unknown positive multiplier k. Once you know one actual value or the total, you can solve for k.

Must Know: For any ratio a:b = m:n, write a = mk and b = nk. Total = (m + n)k. Solve for k using whatever additional information the problem gives you.

Example — The ratio of red to blue marbles is 3:5. There are 40 marbles total. How many are red?

1. Red = 3k, Blue = 5k
2. Total: 3k + 5k = 8k = 40
3. Solve: k = 5
4. Red = 3k = 3 × 5 = 15
5. Verify: 15 + 25 = 40 ✓

6. Rate × Time = Distance (or Work)

The GRE uses the relationship Rate × Time = Distance for motion problems and Rate × Time = Work for job/pipe problems. For combined-work problems, add the individual rates (jobs per hour) and then find the total time.

Must Know: For average speed on a round trip (or any two-leg journey), use total distance / total time — NEVER average the two speeds. Speed averaging is wrong whenever the time spent at each speed differs.

Example — Worker A finishes a job in 3 hours; Worker B in 6 hours. How long together?

1. A's rate = 1/3 job/hr | B's rate = 1/6 job/hr
2. Combined rate = 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2 job/hr
3. Time = 1 job ÷ (1/2 job/hr) = 2 hours
4. Verify: In 2 hours, A does 2/3 of the job and B does 2/6 = 1/3 of the job. Total = 2/3 + 1/3 = 1 ✓

7. Exponent Laws

The GRE tests whether you can apply exponent rules quickly under pressure. Every rule below has exactly one purpose: to let you rewrite expressions into a simpler or comparable form.

Must Know:

• a^m × a^n = a^(m+n) ← same base, multiply → add exponents
• a^m ÷ a^n = a^(m−n) ← same base, divide → subtract exponents
• (a^m)^n = a^(m×n) ← power of a power → multiply exponents
• a^0 = 1 (when a ≠ 0)
• a^(−n) = 1/a^n ← negative exponent flips to denominator
• a^(1/n) = ⁿ√a ← fractional exponent is a root

Example — Simplify: (2^3 × 2^4) / 2^5

1. Numerator: 2^3 × 2^4 = 2^(3+4) = 2^7
2. Divide: 2^7 / 2^5 = 2^(7−5) = 2^2 = 4

8. Absolute Value

Absolute value measures distance from zero — it is always non-negative. The GRE uses absolute value in two main ways: solving equations (two cases) and interpreting inequalities (ranges).

Must Know:

• |x| = k → x = k or x = −k (two solutions)
• |x| < k → −k < x < k (a bounded range — "between")
• |x| > k → x > k or x < −k (two separate rays — "outside")

Example — Solve |2x − 4| = 6:

1. Case 1: 2x − 4 = 6 → 2x = 10 → x = 5
2. Case 2: 2x − 4 = −6 → 2x = −2 → x = −1
3. Solutions: x = 5 or x = −1
4. Verify: |2(5) − 4| = |6| = 6 ✓ | |2(−1) − 4| = |−6| = 6 ✓

Common Traps

• Successive % changes treated as additive: "+20% then −20% = 0%" is wrong. Multiply the multipliers: ×1.20 × ×0.80 = ×0.96. Net result is a 4% loss.
• Wrong base in percent change: Always divide by the OLD value, not the new one. Using the new value gives a different (wrong) percentage.
• Negative exponents misread as negative numbers: 2^(−3) = 1/8, not −8. A negative exponent means "reciprocal," not "negative value."
• Averaging speeds on a round trip: If you drove 60 mph there and 40 mph back, average speed is NOT 50 mph. It's total distance ÷ total time. (Actual answer: 48 mph.)
• Assuming ratio values are actual values: a:b = 3:5 does not mean a = 3. Always introduce the multiplier k.

GRE Strategy

• Convert percents to decimals before multiplying (0.35 × 80 is faster than "35% of 80").
• For successive percent change problems, plug in $100 as your starting value — the arithmetic becomes trivial and you get a concrete net percent.
• When a percent problem has no numbers, plug in 100 as the whole — it turns every percent directly into a number.
• For ratio problems, write out a = mk and b = nk immediately. Don't skip the multiplier step even when it feels obvious.
• For absolute value inequalities, sketch the number line. It takes 5 seconds and eliminates direction errors.

Worked Example

Question: A store buys an item for $200. The store marks up the price by 40%, then later offers a 25% discount on the marked price. What is the final selling price?

(A) $196 (B) $200 (C) $204 (D) $210 (E) $220

Solution:

Step 1 — Apply the 40% markup. Markup multiplier = 1 + 0.40 = 1.40 Marked price = $200 × 1.40 = $280

Step 2 — Apply the 25% discount to the marked price. Discount multiplier = 1 − 0.25 = 0.75 Final price = $280 × 0.75 = $210

Answer: (D) $210

Step 3 — Verify using the combined multiplier. Net multiplier = 1.40 × 0.75 = 1.05 Final price = $200 × 1.05 = $210 ✓ The net effect is a 5% increase over the original price.

Why the wrong answers are wrong:

• (B) $200 is the trap answer for anyone who thinks +40% then −25% nets to +15%, and then separately assumes the store breaks even. This conflates the operations.
• (C) $204 corresponds to a net 2% increase — a result you'd get if you incorrectly averaged 40% and 25% in some way, or applied the discount to the original price instead of the marked price.
• (A) $196 is what you get if you treat the 25% discount as applying to the original $200 (−$50) and then apply only part of the markup — a sign that the problem's two-step structure was missed entirely.

Key takeaway: The 25% discount is applied to the $280 marked price, not the original $200. Successive percent changes always act on the current (modified) value. The multiplier method — 1.40 × 0.75 = 1.05 — is the fastest and most reliable path to the right answer.