**GMAT Focus Edition: Two-Part Analysis – Paired Decisions & Simultaneous Constraints**

GMAT Focus Edition: Two-Part Analysis – Paired Decisions & Simultaneous Constraints

Data Insights Study Guide

What This Is

Two-Part Analysis (TPA) questions on the GMAT Focus Edition test your ability to evaluate paired decisions under simultaneous constraints. You’ll be given a scenario with two interdependent variables (e.g., production quantities, investment allocations, or scheduling choices) and must select one answer from each column that satisfies all given conditions. These questions appear in the Data Insights section and account for ~10-15% of your DI score—mastering them can add 10+ points to your total.

Real-GMAT Example:
A company produces two products, X and Y. Each unit of X requires 2 hours of labor and 1 unit of raw material; each unit of Y requires 1 hour of labor and 3 units of raw material. The company has 100 hours of labor and 120 units of raw material available. If the company must produce at least 10 units of X and at least 5 units of Y, what is the maximum number of units of X and Y that can be produced? (Answer choices are paired: e.g., Column A = 30, Column B = 20.)

Key Concepts & Techniques

1. Simultaneous Constraints (Linear Inequalities)
2. What it is: Two or more inequalities that must be satisfied at the same time (e.g., labor hours + raw materials).
3. When to use: Whenever the problem gives multiple limits (e.g., time, money, resources) that restrict your choices.

4. Pro tip: Graph the inequalities if possible—intersection points are your answer candidates.

5. Paired Decision Variables

6. What it is: Two variables (e.g., X and Y) whose values depend on each other.
7. When to use: When the question asks for two outputs (one from each column) that work together.

8. Pro tip: Assign variables (e.g., x = units of X, y = units of Y) and write equations/inequalities.

9. Feasible Region & Corner Points

10. What it is: The area where all constraints overlap; optimal solutions lie at the corners of this region.
11. When to use: For maximization/minimization problems (e.g., profit, production).

12. Pro tip: Test only the corner points—no need to check every possible value.

13. Integer Constraints

14. What it is: Variables must be whole numbers (e.g., units produced, people assigned).
15. When to use: When the problem specifies discrete quantities (e.g., "units," "people").

16. Pro tip: Round down to the nearest integer if a corner point isn’t whole.

17. Non-Negativity

18. What it is: Variables can’t be negative (e.g., you can’t produce -5 units).
19. When to use: Always—this is a hidden constraint in every TPA question.

20. Pro tip: Add x ≥ 0, y ≥ 0 to your inequalities.

21. Substitution for One Variable

22. What it is: Solve one inequality for y (or x) and substitute into the other.
23. When to use: When one constraint is simple (e.g., x + y = 100).

24. Pro tip: Reduces the problem to a single-variable equation.

25. Testing Answer Choices

26. What it is: Plug in the paired answers to see which satisfies all constraints.
27. When to use: When the problem gives specific pairs (e.g., Column A = 20, Column B = 30).
28. Pro tip: Start with the middle values to eliminate half the options quickly.

Step-by-Step Strategy

Follow these steps for every TPA question:

1. Define Variables
2. Assign letters to the two unknowns (e.g., x = units of X, y = units of Y).

3. Note any minimum/maximum requirements (e.g., x ≥ 10, y ≥ 5).

4. Write Constraints as Inequalities

5. Translate the problem’s limits into math (e.g., 2x + y ≤ 100 for labor).

6. Include non-negativity (x ≥ 0, y ≥ 0).

7. Graph or Solve for Intersection Points

8. Find where the inequalities intersect (e.g., solve 2x + y = 100 and x + 3y = 120).

9. If graphing, plot the lines and identify the feasible region.

10. Test Corner Points

11. Evaluate the objective (e.g., x + y) at each intersection point.

12. For integer constraints, round down to the nearest whole number.

13. Check Answer Choices

14. If the question provides pairs, test them against all constraints.

15. Eliminate pairs that violate any condition.

16. Select the Valid Pair

17. Choose the pair that satisfies all constraints and optimizes the goal (e.g., maximizes production).

Fully Worked Example

Problem:
A bakery makes cakes and pies. Each cake requires 3 cups of flour and 2 eggs; each pie requires 2 cups of flour and 1 egg. The bakery has 60 cups of flour and 40 eggs available. If the bakery must make at least 5 cakes and at least 10 pies, what is the maximum number of cakes and pies it can make? (Answer choices: Column A = Cakes, Column B = Pies)

Step 1: Define Variables
- Let c = number of cakes, p = number of pies.
- Constraints: c ≥ 5, p ≥ 10.

Step 2: Write Inequalities
- Flour: 3c + 2p ≤ 60 - Eggs: 2c + p ≤ 40 - Non-negativity: c ≥ 0, p ≥ 0

Step 3: Find Intersection Points
- Solve 3c + 2p = 60 and 2c + p = 40: - From the second equation: p = 40 – 2c.
- Substitute into the first: 3c + 2(40 – 2c) = 60 → 3c + 80 – 4c = 60 → -c = -20 → c = 20.
- Then p = 40 – 2(20) = 0. But p ≥ 10, so this point is invalid.
- Check other corners: - c = 5 (minimum cakes): 3(5) + 2p = 60 → 2p = 45 → p = 22.5 (round down to 22).
- Check eggs: 2(5) + 22 = 32 ≤ 40 (valid).
- p = 10 (minimum pies): 3c + 2(10) = 60 → 3c = 40 → c ≈ 13.33 (round down to 13).
- Check eggs: 2(13) + 10 = 36 ≤ 40 (valid).

Step 4: Test Objective (Maximize c + p)
- At (c, p) = (5, 22): 5 + 22 = 27 - At (13, 10): 13 + 10 = 23 - Check another point: c = 10, p = 15: - Flour: 3(10) + 2(15) = 60 (valid).
- Eggs: 2(10) + 15 = 35 ≤ 40 (valid).
- c + p = 25 (better than 23 or 27? Wait—27 is higher, but check constraints again).
- For (5, 22): 2(5) + 22 = 32 ≤ 40 (valid).
- For (10, 15): 2(10) + 15 = 35 ≤ 40 (valid).
- 27 is the maximum.

Step 5: Check Answer Choices
- The valid pair is c = 5, p = 22.
- If the choices include (5, 22), select it.

Answer: Column A = 5, Column B = 22.

Common Mistakes

1. Ignoring Minimum Requirements
2. Why it happens: Students focus on the main constraints (flour, eggs) and forget c ≥ 5, p ≥ 10.

3. Correct approach: Write all constraints before solving.

4. Rounding Incorrectly

5. Why it happens: Students round up (e.g., p = 22.5 → 23) and violate constraints.

6. Correct approach: Always round down for integer constraints.

7. Testing Only One Constraint

8. Why it happens: Students check flour but forget eggs (or vice versa).

9. Correct approach: Verify all constraints for every candidate pair.

10. Assuming the Intersection Point is Valid

11. Why it happens: The intersection of 3c + 2p = 60 and 2c + p = 40 gives (c, p) = (20, 0), which violates p ≥ 10.

12. Correct approach: Always check if the intersection satisfies all constraints.

13. Overcomplicating with Calculus

14. Why it happens: Students try to use derivatives or advanced math.
15. Correct approach: Stick to algebra and testing corner points.

GMAT Traps & Timing

1. Hidden Integer Constraints
2. Trap: The problem says "units" or "people" but doesn’t explicitly say "integer."

3. Avoid it: Assume variables must be whole numbers unless stated otherwise.

4. Non-Binding Constraints

5. Trap: One constraint (e.g., eggs) is never the limiting factor.

6. Avoid it: After solving, check which constraint is binding (e.g., flour runs out first).

7. Answer Choices That Violate Minimums

8. Trap: A pair like (4, 30) is given, but c ≥ 5.

9. Avoid it: Eliminate any pair that violates minimum/maximum requirements.

10. Time Budget

11. Spend: 2.5–3 minutes per TPA question.
12. If stuck: Test the answer choices—don’t derive from scratch.

Quick Practice

Question:
A farmer plants wheat and corn. Each acre of wheat requires 2 tons of fertilizer and 3 hours of labor; each acre of corn requires 4 tons of fertilizer and 1 hour of labor. The farmer has 100 tons of fertilizer and 90 hours of labor. If the farmer must plant at least 10 acres of wheat and at least 5 acres of corn, what is the maximum number of acres of wheat and corn that can be planted? (Answer choices: Column A = Wheat, Column B = Corn)

Answer: Column A = 20, Column B = 15.
Explanation: The binding constraint is labor (3w + c ≤ 90). At w = 20, c = 30 (but fertilizer: 2(20) + 4(30) = 160 > 100). Adjust to w = 20, c = 15 (fertilizer: 2(20) + 4(15) = 100; labor: 3(20) + 15 = 75 ≤ 90).

Last-Minute Cram Sheet

1. Always define variables (e.g., x = product X, y = product Y).
2. Write all constraints as inequalities (include x ≥ 0, y ≥ 0).
3. Find intersection points of the constraints—these are your answer candidates.
4. Test corner points in the objective (e.g., maximize x + y).
5. Round down for integer constraints (never up).
6. Check all constraints for every candidate pair.
7. Eliminate answer choices that violate minimums/maximums.
8. Graph if stuck—the feasible region’s corners are the solutions.
9. Non-binding constraints (e.g., extra labor) don’t limit the answer.
10. ⚠️ Trap: The intersection of two constraints may violate a third constraint.