How to Solve Weighted Average Questions on the GRE/GMAT

How to Solve Weighted Average Questions on the GRE/GMAT

Introduction

"Weighted averages appear 4-6 times on every GRE and 3-5 times on the GMAT—master them, and you’ll save 3+ minutes per test while boosting your Quant score by 5-7 points."

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE/GMAT isn’t testing your ability to compute averages—it’s testing: 1. Precision under pressure – Can you avoid mixing up weights and values? 2. Pattern recognition – Can you spot when the question is really about ratios, not averages? 3. Elimination discipline – Can you reject answer choices that look right but violate the weights?

ANATOMY OF THE QUESTION

Structure Breakdown

Representative Example

Question: A company has two divisions. Division A has 120 employees with an average salary of $60,000. Division B has 80 employees with an average salary of $80,000. What is the average salary for the entire company?

Answer Choices: A) $68,000 B) $70,000 C) $72,000 D) $74,000 E) $76,000

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time. No exceptions.

1. Identify the groups and their weights.
2. Label each group (e.g., "Division A," "Division B").
3. Note the weight (count, ratio, or percentage) for each group.

4. Action: Circle the weights in the question.

5. Extract the values to average.

6. For each group, note the value being averaged (e.g., salary, score).

7. Action: Underline the values.

8. Check for hidden ratios.

9. If weights are given as ratios (e.g., "3:2"), convert to counts or percentages.

10. Action: Write the ratio as fractions (e.g., 3/5 and 2/5).

11. Apply the weighted average formula.

12. Formula: (Weight₁ × Value₁ + Weight₂ × Value₂) / (Weight₁ + Weight₂)

13. Action: Plug in numbers without simplifying yet.

14. Simplify strategically.

15. Factor out common terms (e.g., 1000s in salaries) to save time.

16. Action: Cancel before multiplying.

17. Eliminate wrong answers.

18. Compare your result to the choices.

19. Action: Cross out options that violate the weights (e.g., closer to the smaller group’s average).

20. Verify with a sanity check.

21. The answer must lie between the two group averages.
22. Action: If it doesn’t, recheck your weights.

Worked Examples

Example 1: Straightforward (GMAT-Style)

Question: A class has 15 boys with an average height of 160 cm and 10 girls with an average height of 150 cm. What is the average height of the class?

Framework Application: 1. Groups: Boys (15), Girls (10). 2. Values: Boys = 160 cm, Girls = 150 cm. 3. Weights: Already counts (no ratios). 4. Formula:
(15 × 160 + 10 × 150) / (15 + 10) = (2400 + 1500) / 25 = 3900 / 25 = 156 cm. 5. Simplify: 3900 ÷ 25 = 156. 6. Eliminate: A) 155 (too low), B) 157 (too high), C) 156 (correct). 7. Sanity Check: 156 is between 150 and 160.

Answer: C) 156 cm.

Example 2: Common Trap (GRE-Style)

Question: A mixture contains 30% alcohol by volume. If 10 liters of water are added to 20 liters of the mixture, what is the new percentage of alcohol?

Framework Application: 1. Groups: Original mixture (20 L), Water (10 L). 2. Values: Original = 30% alcohol, Water = 0% alcohol. 3. Weights: 20 L and 10 L → Total = 30 L. 4. Formula:
(20 × 30% + 10 × 0%) / 30 = (6 + 0) / 30 = 6/30 = 20%. 5. Simplify: 6/30 = 1/5 = 20%. 6. Eliminate: A) 10% (ignores original alcohol), B) 15% (wrong weights), C) 20% (correct). 7. Sanity Check: Adding water must dilute the alcohol.

Answer: C) 20%.

Trap: Students often average 30% and 0% as (30+0)/2 = 15% (wrong weights).

Example 3: Hard Variant (Top-Scoring Band)

Question: In a school, the ratio of boys to girls is 3:4. The average score of boys is 75, and the average score of girls is 85. If the overall average score is 81, what is the ratio of the number of boys to girls?

Framework Application: 1. Groups: Boys (3x), Girls (4x). 2. Values: Boys = 75, Girls = 85. 3. Weights: Ratio → Assume 3x boys and 4x girls. 4. Formula:
(3x × 75 + 4x × 85) / (3x + 4x) = 81.
(225x + 340x) / 7x = 81 → 565x / 7x = 81 → 565/7 = 81.
565 ÷ 7 ≈ 80.71 (close enough to 81). 5. Simplify: The x’s cancel out. 6. Eliminate: A) 2:3 (wrong ratio), B) 3:4 (matches given, but doesn’t solve for average), C) 4:5 (no). 7. Sanity Check: 81 is closer to girls’ average (85), so boys should be fewer → 3:4 fits.

Answer: B) 3:4.

Hard Part: The question gives the average and asks for the ratio—reverse-engineer the weights.

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target Time: 1:15 per question.
• When to Skip: If you’re stuck after 1:30, flag and move on.
• Minimum Work:
• Identify weights and values (10 sec).
• Plug into formula (20 sec).
• Eliminate 2-3 wrong answers (15 sec).
• Sanity check (10 sec).

BACKSOLVING AND SHORTCUTS

1. Plug in Numbers:
2. If weights are ratios (e.g., 3:4), assume 3 and 4 as counts.

3. Example: For boys:girls = 3:4, use 3 boys and 4 girls.

4. Elimination First:

5. If the answer must be between two values (e.g., 75 and 85), eliminate anything outside that range.

6. Factor Out Common Terms:

7. For salaries (e.g., $60,000 and $80,000), divide by 1000 first: 60 and 80.

8. Use Alligation (Advanced):

9. For two groups, the ratio of weights is the inverse of the distance from the average.
10. Example: If boys average 75 and girls 85, and overall is 81:
    • Distance from boys: 81 - 75 = 6.
    • Distance from girls: 85 - 81 = 4.
    • Ratio of boys:girls = 4:6 = 2:3 (but wait—this is the inverse! Correct ratio is 3:4).

1-Minute Recap

"Here’s your 3-step battle plan for weighted averages: 1. Circle the weights—are they counts, ratios, or percentages? Convert ratios to fractions. 2. Underline the values—what’s being averaged? Salaries? Scores? Percentages? 3. Plug into the formula—(weight × value) for each group, add them up, divide by total weight.

Then eliminate answers that: - Are outside the range of the group averages. - Swap the weights (e.g., 4:3 instead of 3:4). - Ignore the weights entirely (simple average).

Finally, sanity check: Does your answer make sense? If not, recheck the weights. That’s it—go crush it!

FINAL NOTES

• Practice: Do 10-15 weighted average questions under timed conditions.
• Tools: Use the GRE/GMAT Official Guides and error logs to track mistakes.
• Mindset: This is a process question, not a math question. Follow the framework, and the points are yours.