**GMAT Focus Edition: Arithmetic – Ratios, Fractions, Percentages**

GMAT Focus Edition: Arithmetic – Ratios, Fractions, Percentages

Complete Study Guide for High-Scoring Candidates

What This Is

Ratios, fractions, and percentages are the core language of GMAT arithmetic—tested in ~20% of Quant questions (Problem Solving and Data Sufficiency). Mastery here directly impacts your speed, accuracy, and ability to handle multi-step word problems (e.g., mixtures, work rates, profit/loss). Weakness in these concepts leads to careless errors, misinterpreted statements, and wasted time—costing you 50+ points on test day.

Real-GMAT Example:
A store sells two types of shirts: Type A and Type B. The ratio of Type A shirts to Type B shirts is 3:5. If 20% of Type A shirts and 10% of Type B shirts are defective, what percentage of the total shirts are defective? (Answer: ~13.75%)

Key Concepts & Techniques

1. Ratio → Fraction → Percentage Conversion
2. What it is: A ratio (e.g., 3:5) can be written as a fraction (3/8 or 5/8) and converted to a percentage (37.5% or 62.5%).

3. When to use: When a problem gives a ratio but asks for a percentage (or vice versa). Always convert to a common denominator or total parts first.

4. Part-to-Part vs. Part-to-Whole

5. What it is: A ratio like 3:5 can describe parts of a whole (3 parts A, 5 parts B, total 8 parts) or two separate quantities (3 A’s for every 5 B’s).

6. When to use: If the problem mentions a "total," it’s part-to-whole. If it compares two distinct groups (e.g., "apples to oranges"), it’s part-to-part.

7. Cross-Multiplication for Proportions

8. What it is: If a/b = c/d, then ad = bc.

9. When to use: When you have two equivalent ratios (e.g., "If 4 workers build 6 walls in 3 days, how many walls can 10 workers build in 5 days?").

10. Percentage Change Formula

11. What it is: % Change = (New – Old) / Old × 100

12. When to use: For "increase/decrease" problems (e.g., "A price rose from $50 to $60. What’s the % increase?").

13. Successive Percentage Changes

14. What it is: Multiply the multipliers (e.g., 20% increase → ×1.20; 10% decrease → ×0.90).

15. When to use: For problems like "A value increases by 20%, then decreases by 10%. What’s the net change?" (Answer: +8%)

16. Mixture Problems (Alligation)

17. What it is: A method to find the ratio of two components in a mixture.

18. When to use: When combining two solutions with different concentrations (e.g., "How much 30% alcohol must be mixed with 50% alcohol to get 40% alcohol?").

19. Work-Rate Ratios

20. What it is: If A works at rate R₁ and B at rate R₂, their combined rate is R₁ + R₂.

21. When to use: For problems like "A can complete a job in 6 hours, B in 4 hours. How long together?" (Answer: 2.4 hours)

22. Data Sufficiency: Plugging in Numbers

23. What it is: Assign values to variables to test sufficiency (e.g., if a ratio is 3:4, assume 3 and 4, then check if the statements hold).
24. When to use: When a problem gives a ratio but no concrete numbers.

Step-by-Step Strategy

Follow this process for every ratio/fraction/percentage problem:

1. Identify the Total
2. If the problem gives a ratio (e.g., 3:5), assume a total (e.g., 8 parts) or assign variables (e.g., 3x and 5x).

3. Exception: If the ratio is part-to-part (no total mentioned), keep it as 3:5.

4. Convert to a Common Base

5. Ratios → Fractions (3:5 = 3/8 and 5/8 of the total).
6. Fractions → Percentages (3/8 = 37.5%).

7. Percentages → Decimals (20% = 0.20).

8. Set Up an Equation or Table

9. For mixture problems: Use Alligation (see example below).
10. For work rates: Create a Rate × Time = Work table.

11. For percentage changes: Use New = Old × (1 ± % Change).

12. Solve for the Unknown

13. Cross-multiply, combine like terms, or plug in numbers.

14. Pro tip: If stuck, test answer choices (especially for Problem Solving).

15. Check Units and Labels

16. Ensure your answer matches what’s asked (e.g., "percentage" vs. "fraction," "ratio" vs. "actual number").

17. Verify with a Quick Estimate

18. Does your answer make sense? (e.g., A 20% increase on 100 should be 120, not 80.)

Fully Worked Example (GMAT-Style)

Problem:
A company’s revenue in 2022 was $120,000. In 2023, revenue increased by 25%, but expenses increased by 40%. If expenses were 60% of revenue in 2022, what is the ratio of profit in 2023 to profit in 2022?

Solution (Using the Strategy):

1. Identify the Total
2. 2022 Revenue = $120,000.
3. 2022 Expenses = 60% of revenue = 0.60 × $120,000 = $72,000.

4. 2022 Profit = Revenue – Expenses = $120,000 – $72,000 = $48,000.

5. Convert to a Common Base

6. 2023 Revenue = $120,000 × 1.25 = $150,000.
7. 2023 Expenses = $72,000 × 1.40 = $100,800.

8. 2023 Profit = $150,000 – $100,800 = $49,200.

9. Set Up the Ratio

10. Profit 2023 : Profit 2022 = $49,200 : $48,000.

11. Simplify: Divide both by $1,200 → 41:40.

12. Check Units

13. The question asks for a ratio, so 41:40 is correct.

Answer: 41:40

Common Mistakes

1. Mistake: Misinterpreting part-to-part vs. part-to-whole ratios.
2. Why it happens: Students assume a ratio like 3:5 refers to the total (8 parts) when it might not.

3. Correct approach: Ask: "Does the problem mention a total?" If not, keep the ratio as 3:5.

4. Mistake: Adding percentages directly (e.g., 20% + 10% = 30%).

5. Why it happens: Confusing successive changes with simple addition.

6. Correct approach: Multiply the multipliers: 1.20 × 0.90 = 1.08 → 8% net increase.

7. Mistake: Forgetting to convert percentages to decimals.

8. Why it happens: Carelessness (e.g., writing 20% as 20 instead of 0.20).

9. Correct approach: Always convert % → decimal before calculations.

10. Mistake: Solving for the wrong variable in Data Sufficiency.

11. Why it happens: Over-solving instead of testing sufficiency.

12. Correct approach: Stop once you know if the statements are sufficient—don’t calculate the actual value.

13. Mistake: Ignoring units in word problems.

14. Why it happens: Rushing and missing labels (e.g., "dollars" vs. "cents").
15. Correct approach: Circle the units in the question and ensure your answer matches.

GMAT Traps & Timing

1. Trap: Hidden Totals
2. Example: "The ratio of boys to girls is 3:4." → Total students = 7 parts, not 3 or 4.

3. How to avoid: Always ask: "Is this part-to-part or part-to-whole?"

4. Trap: Percentage of What?

5. Example: "A price increased by 20%, then decreased by 20%." → The decrease is on the new price, not the original.

6. How to avoid: Track the base for each percentage change.

7. Trap: Non-Integer Ratios

8. Example: "The ratio of A to B is 1:1.5." → Convert to integers (2:3) to simplify.

9. How to avoid: Multiply ratios by 2, 10, etc., to eliminate decimals.

10. Timing:

11. Problem Solving: 1.5–2 minutes per question.
12. Data Sufficiency: 1–1.5 minutes per question.
13. Pro tip: If stuck, guess and move on—don’t let one question eat 3+ minutes.

Quick Practice

1. Problem:
   A mixture contains alcohol and water in the ratio 5:3. If 4 liters of water is added, the ratio becomes 5:4. What is the original quantity of alcohol?
   Answer: 20 liters.
   Solution Path: Let alcohol = 5x, water = 3x. New water = 3x + 4. Set up 5x/(3x + 4) = 5/4 → x = 4 → Alcohol = 5x = 20.

2. Problem:
   If a number is increased by 25% and then decreased by 20%, what is the net percentage change?
   Answer: 0% (no change).
   Solution Path: 1.25 × 0.80 = 1.00 → 0% change.

Last-Minute Cram Sheet

1. Ratio → Fraction: 3:5 = 3/8 and 5/8 of the total.
2. Percentage Change: (New – Old)/Old × 100.
3. Successive % Changes: Multiply multipliers (e.g., 1.20 × 0.90 = 1.08).
4. Alligation: For mixtures, subtract diagonally to find the ratio.
5. Work Rates: Combined rate = R₁ + R₂.
6. Data Sufficiency: Plug in numbers to test sufficiency.
7. Part-to-Part vs. Part-to-Whole: Check if a total is given.
8. Trap: % of what? Always track the base.
9. Trap: Non-integer ratios → Multiply to eliminate decimals.
10. Time Budget: 1.5–2 min per Problem Solving, 1–1.5 min per DS.

⚠️ Final Tip: On test day, write down the ratio/fraction/percentage conversion before solving—this avoids 90% of careless errors.