GRE Prep: Arithmetic (Number Properties, Percents, Ratios, Exponents & Roots)

GRE – Arithmetic (Number Properties, Percents, Ratios, Exponents & Roots)

GRE Study Guide – Arithmetic (Number Properties, Percents, Ratios, Exponents & Roots)

What This Is (1 short paragraph)

Arithmetic on the GRE tests your ability to manipulate whole numbers, fractions, decimals, percents, ratios, and powers quickly and accurately. These questions appear in both Quantitative Comparison and Multiple‑Choice formats, and they often serve as “warm‑up” items that set the pacing for the rest of the section. For example, a typical stem might read: “If a shirt originally costs \$48 and is discounted by 25 %, what is the sale price?” – a straightforward percent‑of‑a‑whole problem that still requires careful handling of fractions and decimals under time pressure.

Key Terms & Rules (8–12 bullets)

• Whole‑Number Properties: Even/odd, divisibility (2, 3, 5, 9, 10), prime vs. composite; e.g., a number ending in 0 or 5 is divisible by 5.
• Fraction‑to‑Decimal Conversion: Divide numerator by denominator; remember that terminating decimals end when the denominator’s prime factors are only 2 and 5.
• Percent Formula: Percent × Whole = Part → Part = Percent × Whole / 100.
• Ratio Simplification: Reduce by the greatest common divisor (GCD); e.g., 24:36 simplifies to 2:3.
• Proportion Cross‑Multiplication: If a:b = c:d, then a·d = b·c.
• Exponent Rules:
• Product of Powers: a^m · a^n = a^(m+n)
• Power of a Power: (a^m)^n = a^(m·n)
• Negative Exponent: a^(–n) = 1/a^n
• Zero Exponent: a^0 = 1 (a ≠ 0).
• Square‑Root & Cube‑Root Properties: √(a·b) = √a · √b; ³√(a^3) = a.
• Common‑Base Conversion: Write numbers with the same base to compare exponents (e.g., 2^5 vs. 4^? → 4 = 2^2, so 4^x = 2^(2x)).
• Percent Change: New – Old / Old × 100%. Positive for increase, negative for decrease.
• Compound Percent (Repeated Multiplication): After n identical percent changes, multiply by (1 ± p)^n.
• Data‑Sufficiency Trick: When a statement gives a direct numeric relationship (e.g., “x = y”), you can often answer the comparison without solving for the actual values.

Step‑by‑Step / Process Flow (3–6 steps)

1. Read the stem and identify the operation type – percent, ratio, exponent, or root.
2. Convert all quantities to a common format (e.g., turn percents into decimals or fractions; express ratios as fractions).
3. Apply the relevant rule – use the percent formula, simplify the ratio, or invoke exponent laws.
4. Perform arithmetic with mental shortcuts (e.g., split a 25 % discount into 20 % + 5 % or use the GCD to reduce fractions).
5. Check answer‑choice ranges – eliminate any choice that violates basic constraints (negative price, >100 % increase, non‑integer where an integer is required).
6. Verify quickly – plug the answer back into the original statement to ensure consistency.

Common Mistakes (3–5)

• Mistake: Treating a percent as a whole number (e.g., using 25 instead of 0.25).
  Correction: Always divide the percent by 100 before multiplying; 25 % = 0.25.

• Mistake: Forgetting to simplify ratios before cross‑multiplying, leading to large numbers and arithmetic errors.
  Correction: Reduce the ratio to its lowest terms first; 48:72 → 2:3, then work with smaller numbers.

• Mistake: Misapplying exponent rules, such as adding exponents when the bases differ (e.g., 2^3 · 3^2).
  Correction: Only combine exponents when the bases are identical; otherwise, compute each power separately.

• Mistake: Assuming a “percent increase” is the same as a “percent of the original” (e.g., adding 20 % to 50 and thinking the result is 70).
  Correction: Use the percent‑change formula: New = Old × (1 + p).

• Mistake: Ignoring the possibility of multiple valid solutions in a Data‑Sufficiency question and picking “insufficient” too quickly.
  Correction: Test a concrete example that satisfies the given statements; if the comparison holds for all such examples, the statements are sufficient.

Exam Insights (2–4)

1. Most‑tested concept: Converting between percents, fractions, and decimals. The GRE loves to hide a simple 12 % tip or a 3 : 5 ratio inside a word problem.
2. Tricky distractor: Answer choices that are off by a factor of 10 (e.g., 0.12 vs. 12). This tests whether you remembered to divide by 100.
3. Data‑Sufficiency pattern: One statement often gives a direct relationship (e.g., “x = 2y”), which is enough to answer a comparison; the other statement is a red‑herring.
4. Exponent/root shortcut: When the exponent is a multiple of 2 or 3, rewrite the base as a perfect square or cube to simplify (e.g., 8^4 = (2^3)^4 = 2^12).

Quick Check Questions (2–3)

1. Quantitative Comparison
   Statement: Compare A = 30 % of 200 with B = 1/4 of 180.
   Answer: A = B (Choice C).
   Explanation: 30 % of 200 = 0.30 × 200 = 60; 1/4 of 180 = 45; actually 60 > 45 → A > B (Choice A). (Correct answer: A)

2. Multiple‑Choice – Exponents
   
   If 2^x = 8, what is x?
   Answer: 3 (Choice B).
   Explanation: 8 = 2^3, so x = 3.

3. Data‑Sufficiency
   Question: Is the ratio of p to q greater than 2?
   
   1) p = 6, q = 2.
   
   2) p = 4q.
   Answer: A – Statement 1 alone is sufficient; 6/2 = 3 > 2.
   Explanation: Statement 2 only tells us p = 4q, which could be 4/1 = 4 (>2) but also 8/4 = 2 (not >2); insufficient alone, but we already have sufficiency from statement 1.

Last‑Minute Cram Sheet (10 one‑liners)

• ⚠️ Percent → Decimal: Always divide by 100 (25 % = 0.25).
• Divisibility Rule: A number is divisible by 3 if the sum of its digits is a multiple of 3.
• Ratio → Fraction: a:b = a/b; simplify using GCD.
• Exponent Shortcut: (a^m)^n = a^(m·n).
• Negative Exponent: a^(–n) = 1/a^n.
• Zero Exponent: a^0 = 1 (a ≠ 0).
• Square‑Root of a Square: √(a^2) = |a| (absolute value).
• Compound Percent: After n identical 10 % increases, multiply by (1.10)^n.
• Cross‑Multiplication: For a/b = c/d, check a·d = b·c.
• ⚠️ Data‑Sufficiency Trap: Never assume a variable’s value; test with a concrete example that satisfies the given statements.