**GRE Advanced Mixture Problems: Weighted Averages & Alligation Method**

GRE Advanced Mixture Problems: Weighted Averages & Alligation Method

Mastering this topic can add 2–4 points to your Quant score by turning seemingly complex word problems into fast, mechanical calculations.

What This Is

Mixture problems test your ability to combine quantities with different properties (e.g., concentrations, prices, speeds) and find a weighted average. The GRE loves these because they disguise algebra in real-world scenarios—think solutions, alloys, or investment portfolios. A typical question:

"A chemist mixes 5 liters of a 20% saline solution with 3 liters of a 40% saline solution. What is the concentration of the final mixture?"

Mastering weighted averages and the alligation method (a shortcut for mixture problems) lets you solve these in under 60 seconds, freeing up time for harder questions.

Key Concepts & Techniques

1. Weighted Average Formula
2. What it is: The average of two (or more) groups, where each group contributes proportionally to its size.
3. When to use: Any problem where two quantities are combined (e.g., solutions, investments, grades).

4. Formula:
   [
   \text{Weighted Average} = \frac{(Q_1 \times V_1) + (Q_2 \times V_2)}{V_1 + V_2}
   ]
   where (Q) = quantity (e.g., concentration, price) and (V) = volume/weight.

5. Alligation Method (The "X" Diagram)

6. What it is: A visual shortcut to find the ratio of two components in a mixture without algebra.
7. When to use: When you need the ratio of two solutions to achieve a target concentration (or any weighted average).

8. How it works:

   
   
   1. Write the two given quantities on the left (e.g., 20% and 40%).
   2. Write the target quantity in the middle (e.g., 30%).
   3. Subtract diagonally to find the ratio of the parts:
      [
      \begin{array}{ccc}
      20 & \quad & 40 \
      & 30 & \
      \end{array}
      ]
      → (40 - 30 = 10) (parts of 20%)
      → (30 - 20 = 10) (parts of 40%)
      → Ratio = 1:1 (equal parts of each solution).

9. Total Quantity vs. Component Quantity

10. What it is: Distinguishing between the total mixture (e.g., 8 liters) and the amount of the key substance (e.g., 2.4 liters of salt).

11. When to use: When the question asks for the amount of a substance (not just concentration).

12. Three-Component Mixtures

13. What it is: Extending weighted averages to three groups (e.g., three solutions mixed together).

14. When to use: Rare but tested; use the weighted average formula or alligation twice (combine two first, then add the third).

15. Price Mixtures (Cost per Unit)

16. What it is: Problems where two items (e.g., nuts, coffees) are mixed to achieve a target price per pound.

17. When to use: When the "quantity" is price, and the "volume" is weight/units.

18. Replacement Problems

19. What it is: Removing some of a mixture and replacing it with another substance (e.g., draining and refilling a tank).
20. When to use: When the problem describes a process (e.g., "a chemist removes 2 liters and adds 2 liters of water").
21. Key insight: The amount removed is proportional to the original mixture.

Step-by-Step Strategy

Follow these steps for every mixture problem:

1. Identify the components and target.
2. What are the two (or more) quantities being mixed? (e.g., 20% and 40% solutions)

3. What is the target? (e.g., 30% concentration, $5 per pound)

4. Decide: Weighted Average or Alligation?

5. Use weighted average if you need the final value (e.g., "What is the concentration?").

6. Use alligation if you need the ratio of the components (e.g., "In what ratio should they be mixed?").

7. Set up the equation or diagram.

8. For weighted average: Plug into the formula.

9. For alligation: Draw the "X" and subtract diagonally.

10. Solve for the unknown.

11. If given a ratio, scale it to the actual quantities.

12. If given quantities, calculate the weighted average.

13. Check units and answer the question.

14. Did you find the concentration, the amount of substance, or the ratio? Match the answer choice.

Fully Worked Example (GRE-Style)

Question:
A grocer mixes almonds that cost $8 per pound with cashews that cost $12 per pound. If the grocer wants a 10-pound mixture that costs $10 per pound, how many pounds of almonds should be used?

Solution:

1. Identify components and target:
2. Almonds: $8/lb
3. Cashews: $12/lb

4. Target: 10 lbs at $10/lb

5. Decide: Alligation (we need the ratio).

6. Draw the "X":
   [
   \begin{array}{ccc}
   8 & \quad & 12 \
   & 10 & \
   \end{array}
   ]
7. Subtract diagonally:
   • (12 - 10 = 2) (parts of $8 almonds)
   • (10 - 8 = 2) (parts of $12 cashews)

8. Ratio = 2:2 or 1:1.

9. Scale to total quantity:

10. Total parts = 1 + 1 = 2.
11. Total mixture = 10 lbs.

12. Almonds = ((1/2) \times 10 = 5) lbs.

13. Check:

14. Cost = ((5 \times 8) + (5 \times 12) = 40 + 60 = 100).
15. Total pounds = 10 → $100/10 lbs = $10/lb. ✔️

Answer: 5 pounds of almonds.

Common Mistakes

1. Mistake: Mixing up the ratio from alligation.
2. Why it happens: Students subtract the wrong diagonal (e.g., (8 - 10) instead of (12 - 10)).

3. Correct approach: Always subtract the larger number minus the target and target minus the smaller number.

4. Mistake: Forgetting to scale the ratio to actual quantities.

5. Why it happens: The alligation gives a ratio (e.g., 1:2), but the question asks for pounds/liters.

6. Correct approach: Multiply the ratio by the total quantity to get actual amounts.

7. Mistake: Using the wrong formula for replacement problems.

8. Why it happens: Students treat replacement as a simple mixture, ignoring that some of the original is removed.

9. Correct approach: Calculate the remaining original mixture first, then add the new substance.

10. Mistake: Misinterpreting "concentration" vs. "amount."

11. Why it happens: The question asks for the amount of salt (e.g., 2 liters), but the student calculates the concentration (e.g., 25%).
12. Correct approach: Read carefully: "What is the concentration?" vs. "How much salt is in the mixture?"

GRE Traps & Timing

1. Trap: Hidden Ratios
2. The GRE may give you a ratio (e.g., "mixed in a 2:3 ratio") but ask for the total quantity or final concentration.

3. Avoid it: Always confirm what the question is asking—don’t assume it’s about the ratio.

4. Trap: Replacement Problems

5. Example: "A 20-liter solution is 30% alcohol. If 5 liters are removed and replaced with water, what is the new concentration?"

6. Avoid it: Calculate the remaining alcohol first: (20 \times 0.3 = 6) liters → remove 5 liters (30% of 5 = 1.5 liters alcohol removed) → remaining alcohol = (6 - 1.5 = 4.5) liters.

7. Timing:

8. Weighted average problems: 60–90 seconds.
9. Alligation problems: 45–60 seconds.
10. Replacement problems: 90–120 seconds (more steps).

Quick Practice

1. Question:
   A 12-liter solution is 25% acid. How many liters of a 50% acid solution must be added to make a 30% acid solution?
   Answer: 3 liters
   Solution Path: Use alligation to find the ratio (25% and 50% to 30% → 20:5 or 4:1). Total parts = 5, so 1 part = 3 liters.

2. Question:
   Coffee A costs $6 per pound and Coffee B costs $9 per pound. If 4 pounds of Coffee A are mixed with 6 pounds of Coffee B, what is the cost per pound of the mixture?
   Answer: $7.80
   Solution Path: Weighted average: ((4 \times 6 + 6 \times 9)/10 = (24 + 54)/10 = 78/10 = 7.8).

Last-Minute Cram Sheet

1. Weighted Average Formula: (\frac{(Q_1 \times V_1) + (Q_2 \times V_2)}{V_1 + V_2}).
2. Alligation: Subtract diagonally to get the ratio (larger - target, target - smaller).
3. Replacement Problems: Calculate remaining substance before adding the new one.
4. Ratio → Quantity: Multiply the ratio by the total to get actual amounts.
5. Concentration vs. Amount: "25% solution" ≠ "25 liters of acid."
6. Three-Component Mixtures: Combine two first, then add the third.
7. Price Mixtures: Treat cost per unit as the "quantity" in the formula.
8. Trap: The GRE may give a ratio but ask for the total quantity.
9. Trap: Replacement problems remove proportional amounts of the original mixture.
10. Time-Saver: Alligation is faster than algebra for ratio questions.