**GMAT Focus Edition: Arithmetic Word Problems – Rates, Work, Mixtures**

GMAT Focus Edition: Arithmetic Word Problems – Rates, Work, Mixtures

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What This Is

The GMAT tests arithmetic word problems involving rates, work, and mixtures because they measure your ability to translate real-world scenarios into mathematical relationships—a skill critical for business school and beyond. These problems assess logical reasoning, algebraic manipulation, and attention to units, often appearing in Problem Solving (PS) and Data Sufficiency (DS) formats.

Example (GMAT-Style):
Working together, Machine A and Machine B can complete a job in 6 hours. Machine A alone takes 10 hours to complete the job. How long does it take Machine B alone to complete the job? (A) 12 hours (B) 15 hours (C) 18 hours (D) 20 hours (E) 24 hours

Mastering these problems can boost your Quant score by 30–50 points by eliminating careless errors and improving speed.

Key Concepts & Techniques

1. Rate × Time = Work (RTW) Framework
2. What it is: The core equation for work problems: Rate × Time = Work. Rate is work per unit time (e.g., jobs/hour).
3. When to use: Any problem involving multiple workers, machines, or pipes completing a task.

4. Pro tip: Always express rates in the same units (e.g., jobs/hour, not jobs/minute).

5. Combined Rates (Additive for Parallel Work)

6. What it is: If two workers work together, their rates add: R₁ + R₂ = R_combined.
7. When to use: Problems where entities work simultaneously (e.g., two pipes filling a tank).

8. Example: If A’s rate = 1/10 job/hour and B’s rate = 1/15 job/hour, combined rate = 1/10 + 1/15 = 1/6 job/hour.

9. Relative Rates (Subtractive for Opposing Work)

10. What it is: If two entities work against each other (e.g., one fills a tank, another drains it), subtract rates: R₁ – R₂ = R_net.

11. When to use: Problems with opposing actions (e.g., filling vs. draining, upstream vs. downstream travel).

12. Mixture Problems: Weighted Averages

13. What it is: The final concentration is the weighted average of the components.
14. Formula: (Amount₁ × Concentration₁ + Amount₂ × Concentration₂) / (Amount₁ + Amount₂) = Final Concentration.

15. When to use: Problems involving combining solutions, alloys, or investments with different percentages.

16. Distance = Rate × Time (DRT) Table

17. What it is: A table organizing Distance, Rate, Time for each traveler or leg of a trip.
18. When to use: Motion problems (e.g., two cars traveling toward each other, round trips).

19. Example:
    | Traveler | Distance | Rate | Time |
    |----------|----------|------|------|
    | Car A | D | 60 | T |
    | Car B | D | 40 | T |

20. Setting Up Equations from Words

21. What it is: Translate phrases like "twice as fast" or "30% more" into algebraic expressions.
22. When to use: All word problems—this is the first step before solving.

23. Example: "Machine B is 20% faster than Machine A" → R_B = 1.2 × R_A.

24. Sufficiency Logic (DS-Specific)

25. What it is: For Data Sufficiency, never solve—just determine if the statements provide enough information to find a unique answer.

26. When to use: All DS problems—focus on relationships, not calculations.

27. Unit Consistency

28. What it is: Ensure all units (hours, minutes, miles, km) are consistent before solving.
29. When to use: Every problem—mixing units (e.g., hours and minutes) is a top cause of errors.

Step-by-Step Strategy

Follow this 6-step process for every rate/work/mixture problem:

1. Identify the Type
2. Is it work (machines, people), motion (distance/rate/time), or mixture (solutions, investments)?

3. Example: "Two pipes fill a tank" → Work problem.

4. Define Variables & Units

5. Assign variables to unknowns (e.g., R_A = rate of Machine A in jobs/hour).

6. Write down units (e.g., "R_A = 1/10 job/hour").

7. Set Up the Equation(s)

8. For work problems, use Rate × Time = Work.
9. For mixtures, use weighted averages.

10. For motion, use Distance = Rate × Time and create a table.

11. Solve for the Unknown

12. Isolate the variable using algebra (e.g., cross-multiply, combine like terms).

13. Check units at each step.

14. Verify the Answer

15. Plug the answer back into the problem to ensure it makes sense.

16. Example: If Machine B takes 15 hours alone, does the combined rate match the given 6 hours?

17. Eliminate Trap Answers

18. Common traps: inverted rates (e.g., 10 hours vs. 1/10 job/hour), unit mismatches, or partial answers (e.g., time for one worker when the question asks for both).

Fully Worked Example (Using the Strategy)

Problem:
Working together, Machine A and Machine B can complete a job in 6 hours. Machine A alone takes 10 hours to complete the job. How long does it take Machine B alone to complete the job? (A) 12 hours (B) 15 hours (C) 18 hours (D) 20 hours (E) 24 hours

Step 1: Identify the Type
→ Work problem (machines completing a job).

Step 2: Define Variables & Units
- Let R_A = rate of Machine A (jobs/hour).
- Let R_B = rate of Machine B (jobs/hour).
- Given: Machine A alone takes 10 hours → R_A = 1/10 job/hour.
- Combined rate: R_A + R_B = 1/6 job/hour (since together they take 6 hours).

Step 3: Set Up the Equation
R_A + R_B = 1/6 1/10 + R_B = 1/6

Step 4: Solve for R_B
R_B = 1/6 – 1/10 R_B = (5 – 3)/30 = 2/30 = 1/15 job/hour

Step 5: Find Time for Machine B Alone
- Rate = 1/15 job/hour → Time = 15 hours (since Time = 1/Rate).

Step 6: Verify & Eliminate Traps
- Combined rate: 1/10 + 1/15 = 1/6 (matches given).
- Trap answer: (A) 12 hours is R_A + 3 hours—irrelevant.

Answer: (B) 15 hours

Common Mistakes

1. Mistake: Confusing rate and time.
2. Why it happens: Students invert the relationship (e.g., writing R = Time/Work instead of R = Work/Time).

3. Correct approach: Always write Rate = Work/Time (e.g., jobs/hour).

4. Mistake: Adding times instead of rates for combined work.

5. Why it happens: Misapplying the idea of "working together" (e.g., adding 10 hours + 15 hours = 25 hours).

6. Correct approach: Rates add, times do not. Use R_A + R_B = R_combined.

7. Mistake: Ignoring units (e.g., mixing hours and minutes).

8. Why it happens: Carelessness under time pressure.

9. Correct approach: Convert all units before setting up equations (e.g., 30 minutes = 0.5 hours).

10. Mistake: Misinterpreting mixture problems as simple averages.

11. Why it happens: Assuming equal quantities when the problem states otherwise.

12. Correct approach: Use weighted averages (e.g., 30% of 100g + 50% of 200g ≠ 40% of 300g).

13. Mistake: Solving Data Sufficiency problems completely.

14. Why it happens: Overcomplicating—DS only requires sufficiency, not the answer.
15. Correct approach: Ask: "Can I find a unique value with this statement?" If yes, it’s sufficient.

GMAT Traps & Timing

1. Trap: Inverted Rates
2. What it is: Answer choices include 1/rate (e.g., if the answer is 15 hours, 1/15 job/hour may appear).

3. How to avoid: Always write units (e.g., "R_B = 1/15 job/hour" → time = 15 hours).

4. Trap: Partial Answers

5. What it is: The question asks for combined time, but the answer is for one worker.

6. How to avoid: Reread the question before selecting an answer.

7. Trap: Opposing Work (Filling vs. Draining)

8. What it is: Problems where one entity adds and another subtracts (e.g., a pipe fills while a drain empties).

9. How to avoid: Subtract rates (e.g., R_fill – R_drain = R_net).

10. Timing Budget

11. Problem Solving: 2–2.5 minutes per question.
12. Data Sufficiency: 1.5–2 minutes per question (spend less time on calculations).

Quick Practice

Question 1:
Two pipes, A and B, can fill a tank in 4 hours and 6 hours respectively. If both pipes are opened together, how long will it take to fill the tank? (A) 2 hours (B) 2.4 hours (C) 3 hours (D) 3.6 hours (E) 5 hours

Answer: (B) 2.4 hours
Solution Path: Combined rate = 1/4 + 1/6 = 5/12 tank/hour → Time = 12/5 = 2.4 hours.

Question 2 (DS):
A solution is made by mixing Solution X (20% alcohol) and Solution Y (50% alcohol). Is the resulting mixture 30% alcohol? (1) The ratio of Solution X to Solution Y is 2:1. (2) The total volume of the mixture is 300 mL.

Answer: (A) Statement (1) alone is sufficient.
Solution Path: (1) gives the ratio → weighted average can be calculated. (2) gives volume but not ratio → insufficient.

Last-Minute Cram Sheet

1. Work Problems: Rate × Time = Work → Rates add for parallel work, subtract for opposing work.
2. Motion Problems: Distance = Rate × Time → Create a DRT table.
3. Mixtures: Weighted average = (Amount₁ × %₁ + Amount₂ × %₂) / (Amount₁ + Amount₂).
4. Combined Rates: 1/T_combined = 1/T_A + 1/T_B (for two workers).
5. Opposing Rates: R_net = R_fill – R_drain.
6. Units Matter: Convert all times to hours or minutes—never mix.
7. DS Shortcut: Never solve—just check if the statements uniquely determine the answer.
8. Trap Answer: If the answer is 15 hours, 1/15 may appear—write units!
9. Partial Answers: The question asks for combined time—don’t pick the time for one worker.
10. Time Budget: PS = 2–2.5 min, DS = 1.5–2 min.

Final Tip: On test day, write down the equation first—this forces clarity and reduces errors. Good luck!