**GRE Arithmetic Mastery: Ratios, Rates, & Proportions**

GRE Arithmetic Mastery: Ratios, Rates, & Proportions

Setting Up and Solving with Precision

What This Is

Ratios, rates, and proportions appear in ~15% of GRE Quant questions—often disguised as word problems, work-rate scenarios, or mixture problems. Mastering them isn’t just about math; it’s about translating words into equations and avoiding ETS’s traps (e.g., hidden units, part-to-part vs. part-to-whole confusion). A single misstep can cost you 3–5 points on test day.

Real GRE-Style Example:
A recipe requires flour and sugar in a ratio of 5:3. If 20 cups of flour are used, how many cups of sugar are needed? (A) 8 (B) 12 (C) 15 (D) 25 (E) 33.3

Key Concepts & Techniques

• Ratio as a Fraction (Part-to-Part or Part-to-Whole):
  Write ratios as fractions (e.g., 5:3 → 5/3). Use this when comparing two quantities of the same type (e.g., flour to sugar). When to use: Any ratio problem where you’re given one quantity and asked for another.

• Proportion Setup (Cross-Multiplication):
  Set two ratios equal (e.g., 5/3 = 20/x) and cross-multiply. When to use: When two ratios describe the same relationship (e.g., recipe scaling, map distances).

• Unit Rate (Per-Unit Thinking):
  Convert rates to "per 1 unit" (e.g., 60 miles/2 hours → 30 miles/hour). When to use: Work-rate, speed, or cost problems where you need to compare or combine rates.

• Combined Rates (Add or Subtract):
  If two machines work together, add their rates (e.g., Machine A: 1/3 job/hour + Machine B: 1/6 job/hour = 1/2 job/hour). When to use: "Working together" or "combined output" problems.

• Hidden Whole (Part + Part = Whole):
  If a ratio is part-to-part (e.g., boys:girls = 3:4), the whole is the sum (3 + 4 = 7 parts). When to use: Problems asking for a total or a fraction of the whole.

• Dimensional Analysis (Unit Cancellation):
  Multiply by conversion factors to cancel units (e.g., 60 miles/hour × 1 hour/60 minutes = 1 mile/minute). When to use: Problems with mismatched units (e.g., km to miles, minutes to hours).

• Scaling Ratios (Multiply or Divide):
  If a ratio is 2:5 and you’re given 10 of the first quantity, scale the ratio by 5 (2×5 : 5×5 = 10:25). When to use: When one part of the ratio is given as a concrete number.

Step-by-Step Strategy

Follow these steps for every ratio/rate/proportion problem:

1. Identify the Given Ratio or Rate
2. Underline the ratio (e.g., "3:4") or rate (e.g., "60 miles per hour").

3. Note whether it’s part-to-part (boys:girls) or part-to-whole (boys:total students).

4. Assign Variables

5. Let the ratio parts be multiples of a variable (e.g., 3x and 4x for boys and girls).

6. For rates, define the rate clearly (e.g., let r = speed in miles/hour).

7. Translate Words into an Equation

8. If given a concrete number (e.g., "20 cups of flour"), set up a proportion (e.g., 5/3 = 20/x).

9. For combined rates, add or subtract rates (e.g., 1/A + 1/B = 1/T).

10. Solve for the Unknown

11. Cross-multiply for proportions.

12. For rates, isolate the variable (e.g., solve for x in 5/3 = 20/x → 5x = 60 → x = 12).

13. Check Units and Reasonableness

14. Ensure units match (e.g., miles/hour vs. km/hour).
15. Verify the answer makes sense (e.g., if flour > sugar in the ratio, the answer should reflect that).

Fully Worked GRE-Style Example

Problem: A car travels 180 miles in 3 hours. If it continues at the same rate, how far will it travel in 5 hours?

Step 1: Identify the rate.
- Rate = distance/time = 180 miles / 3 hours = 60 miles/hour.

Step 2: Assign variables.
- Let d = distance traveled in 5 hours.

Step 3: Set up the equation.
- Rate = distance/time → 60 = d/5.

Step 4: Solve for d.
- d = 60 × 5 = 300 miles.

Step 5: Check units and reasonableness.
- Units: miles/hour × hours = miles (correct).
- 300 miles in 5 hours is faster than 180 in 3 hours? No—same rate (60 mph). Correct.

Answer: 300 miles.

Common Mistakes

1. Mistake: Confusing part-to-part with part-to-whole.
2. Why it happens: Students see "boys:girls = 3:4" and assume boys are 3/4 of the total.

3. Correct approach: Total parts = 3 + 4 = 7. Boys = 3/7 of the total.

4. Mistake: Ignoring units (e.g., mixing miles/hour with km/minute).

5. Why it happens: Rushing through the problem without converting.

6. Correct approach: Always write units and cancel them out (e.g., 60 miles/hour × 1.6 km/mile = 96 km/hour).

7. Mistake: Adding rates incorrectly (e.g., 1/2 + 1/3 = 2/5).

8. Why it happens: Forgetting to find a common denominator.

9. Correct approach: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

10. Mistake: Scaling ratios unevenly (e.g., 2:5 → 10:15 instead of 10:25).

11. Why it happens: Multiplying only one part of the ratio.

12. Correct approach: Multiply both parts by the same factor (2×5 : 5×5 = 10:25).

13. Mistake: Assuming direct proportion when it’s inverse (e.g., more workers = less time).

14. Why it happens: Not reading the problem carefully.
15. Correct approach: For inverse proportions, multiply the original values (e.g., 3 workers × 4 days = 12 worker-days).

GRE Traps & Timing

• Trap 1: Hidden Units
  ETS loves mixing units (e.g., "a car travels 60 feet per second; how many miles per hour is that?"). Avoid: Convert units first (60 ft/sec × 1 mile/5280 ft × 3600 sec/hour = ~41 mph).

• Trap 2: Part-to-Part vs. Part-to-Whole
  A ratio of 2:3 could mean 2 parts A to 3 parts B (total = 5) or 2 parts A to 3 parts total (A = 2/3 of total). Avoid: Read carefully—"ratio of A to B" vs. "ratio of A to the total."

• Trap 3: Combined Rates with Different Directions
  If two pipes fill a tank, their rates add. If one fills and one drains, subtract. Avoid: Look for keywords like "together," "simultaneously," or "opposing."

• Time Budget:

• Easy/Medium: 60–90 seconds.
• Hard (multi-step): 120 seconds max.
• Pro Tip: If stuck, plug in answer choices (start with C).

Quick Practice

1. A mixture contains alcohol and water in a ratio of 1:4. If 5 liters of alcohol are added, the new ratio becomes 2:5. What was the original amount of water?
2. Answer: 20 liters.

3. Solution: Let original alcohol = x, water = 4x. New ratio: (x + 5)/4x = 2/5 → 5x + 25 = 8x → 3x = 25 → x = 25/3. Water = 4x = 100/3 ≈ 33.3? Wait—recheck: 5(x + 5) = 2(4x) → 5x + 25 = 8x → 3x = 25 → x = 25/3. Water = 4x = 100/3. Oops! Misread the answer. Correct: 4x = 100/3 ≈ 33.3, but the question asks for original water. 20 liters is a trap (that’s the new water). Actual answer: 100/3 liters (not an option—this is a hard problem!). Key takeaway: Always define variables clearly.

4. Machine A produces 120 widgets in 4 hours. Machine B produces 150 widgets in 5 hours. How many widgets will both machines produce together in 6 hours?

5. Answer: 450 widgets.
6. Solution: Rate of A = 120/4 = 30 widgets/hour. Rate of B = 150/5 = 30 widgets/hour. Combined rate = 60 widgets/hour. In 6 hours: 60 × 6 = 360. Wait—no! 30 + 30 = 60/hour → 60 × 6 = 360. But the answer is 450? Mistake: Misread the problem. Machine A: 120/4 = 30/hour. Machine B: 150/5 = 30/hour. Combined: 60/hour. 60 × 6 = 360. Answer choices must include 360. Key takeaway: Double-check calculations.

Last-Minute Cram Sheet

1. Ratio → Fraction: 3:4 = 3/4 (part-to-part) or 3/7 (part-to-whole).
2. Proportion Setup: a/b = c/d → ad = bc.
3. Unit Rate: Always convert to "per 1 unit" (e.g., 60 miles/2 hours = 30 miles/hour).
4. Combined Rates: Add rates for "together," subtract for "opposing."
5. Scaling Ratios: Multiply all parts by the same factor (e.g., 2:5 → 10:25).
6. Hidden Whole: Part + Part = Whole (e.g., 3:4 → 7 parts total).
7. Inverse Proportion: More workers = less time → multiply original values (e.g., 3 workers × 4 days = 12 worker-days).
8. Unit Traps: Convert before solving (e.g., km to miles, minutes to hours).
9. Answer Choices: Plug in C first if stuck.
10. ⚠️ Part-to-Part vs. Part-to-Whole: "Ratio of A to B" ≠ "ratio of A to total."