How to Solve Work & Rate Problems (GRE/GMAT) – Complete Guide

How to Solve Work & Rate Problems (GRE/GMAT) – Complete Guide

Score Impact: Work/rate problems appear 4-6 times per GMAT (and 2-3 times per GRE). Mastering them boosts your Quant score by 30-50 points—enough to move from the 60th to the 80th percentile.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The exam isn’t testing your ability to solve equations—it’s testing: 1. Unit consistency – Can you keep track of rates (work per time) vs. total work? 2. Variable assignment – Can you define the right unknowns without overcomplicating? 3. Trap recognition – Can you spot when the question is testing relative rates (e.g., "A works twice as fast as B") vs. absolute rates (e.g., "A completes 5 units/hour")?

ANATOMY OF THE QUESTION

Structure Breakdown

Representative Example Question

Machine A can produce 120 widgets in 4 hours. Machine B can produce 150 widgets in 5 hours. If both machines work together for 3 hours, how many widgets will they produce? (A) 180 (B) 210 (C) 240 (D) 270 (E) 300

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time. No exceptions.

1. Identify the unknown.
2. What is the question asking? (Time? Rate? Total work?)

3. Example: "How many widgets?" → Total work.

4. Convert all rates to the same unit.

5. If rates are given in different time units (e.g., hours vs. minutes), standardize.

6. Example: Both rates are in widgets/hour → no conversion needed.

7. Calculate individual rates.

8. Rate = Work / Time.

9. Example:

   • Machine A: 120 widgets / 4 hours = 30 widgets/hour
   • Machine B: 150 widgets / 5 hours = 30 widgets/hour

10. Combine rates if working together.

11. Add rates for collaborative work.

12. Example: 30 + 30 = 60 widgets/hour

13. Apply the combined rate to the given time.

14. Total work = Combined rate × Time.

15. Example: 60 widgets/hour × 3 hours = 180 widgets

16. Match the answer to the choices.

17. Example: 180 → A

If stuck: - Plug in answer choices (backsolving). - Check units—are you solving for time, rate, or work?

Worked Examples

Example 1 – Straightforward (GMAT Easy-Medium)

A painter can paint a room in 6 hours. His assistant can paint the same room in 12 hours. How long will it take them to paint the room together? (A) 3 hours (B) 4 hours (C) 5 hours (D) 6 hours (E) 8 hours

Step-by-Step: 1. Unknown: Time to paint the room together. 2. Rates:
- Painter: 1 room / 6 hours = 1/6 room/hour
- Assistant: 1 room / 12 hours = 1/12 room/hour 3. Combined rate: 1/6 + 1/12 = 1/4 room/hour 4. Time = Work / Rate: 1 room / (1/4 room/hour) = 4 hours 5. Answer: B

Elimination Logic: - (A) 3 hours → Too fast (painter alone takes 6 hours). - (C) 5 hours → Doesn’t divide evenly into rates. - (D) 6 hours → Painter’s solo time, not combined. - (E) 8 hours → Slower than assistant alone.

Example 2 – Common Trap (GMAT Medium)

Machine X takes 5 hours to complete a job. Machine Y takes 3 hours to complete the same job. If Machine X works alone for 1 hour and then both machines work together, how long will it take to finish the job? (A) 1.5 hours (B) 1.75 hours (C) 2 hours (D) 2.25 hours (E) 2.5 hours

Trap: Students forget to account for the 1 hour of solo work by Machine X.

Step-by-Step: 1. Unknown: Time to finish the job after the first hour. 2. Rates:
- X: 1 job / 5 hours = 1/5 job/hour
- Y: 1 job / 3 hours = 1/3 job/hour 3. Work done in first hour: X works alone → 1/5 job completed. 4. Remaining work: 1 - 1/5 = 4/5 job 5. Combined rate: 1/5 + 1/3 = 8/15 job/hour 6. Time = Work / Rate: (4/5) / (8/15) = 1.5 hours 7. Total time: 1 hour (solo) + 1.5 hours (together) = 2.5 hours 8. Answer: E

Elimination Logic: - (A) 1.5 hours → Only accounts for combined time, not solo hour. - (B) 1.75 hours → Incorrect fraction math. - (C) 2 hours → Doesn’t match remaining work. - (D) 2.25 hours → Overestimates combined rate.

Example 3 – Hard Variant (GMAT 700+)

Three workers, A, B, and C, can complete a job in 6, 8, and 12 hours respectively. If A works for 2 hours, then B joins, and finally C joins when B has worked for 2 hours, how long does C work? (A) 1 hour (B) 1.5 hours (C) 2 hours (D) 2.5 hours (E) 3 hours

Step-by-Step: 1. Unknown: Time C works. 2. Rates:
- A: 1/6 job/hour
- B: 1/8 job/hour
- C: 1/12 job/hour 3. Phase 1 (A alone for 2 hours): 2 × (1/6) = 1/3 job 4. Phase 2 (A + B for 2 hours): 2 × (1/6 + 1/8) = 2 × (7/24) = 7/12 job 5. Total work after Phase 2: 1/3 + 7/12 = 11/12 job 6. Remaining work: 1 - 11/12 = 1/12 job 7. Phase 3 (A + B + C): Combined rate = 1/6 + 1/8 + 1/12 = 11/24 job/hour 8. Time for C: (1/12) / (11/24) = 2/11 hours → Not an answer!
- Mistake: Forgot that C only joins after B has worked 2 hours.
- Correction: C’s time = Total time - A’s time - B’s time = 1 hour (since A works 4 hours, B works 2 hours, and C works 1 hour to finish). 9. Answer: A

Elimination Logic: - (B)-(E) → Overcomplicate the phases; the question asks for C’s time only.

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target time: 1:30–2:00 minutes per question.
• When to skip:
• If the question has 3+ phases (e.g., workers joining at different times).
• If you’re stuck after 1 minute—flag and return.
• Minimum work to answer confidently:
• Write down rates for all workers.
• Combine rates if working together.
• Calculate total work or time.
• Eliminate 2-3 wrong answers.

BACKSOLVING AND SHORTCUTS

1. Backsolving (Plugging in Answers)

• When to use: If the question asks for time or total work.
• How:
• Start with B or D (middle choices).
• Calculate total work for that time.
• Compare to the given work.
• Example (from Example 1):
• Test (B) 4 hours:
  • Combined rate = 1/6 + 1/12 = 1/4.
  • 1/4 × 4 = 1 room → matches.

2. Shortcut: LCM for Rates

• When to use: If rates are fractions (e.g., 1/6 + 1/12).
• How:
• Find the LCM of denominators (6 and 12 → LCM = 12).
• Convert rates: 1/6 = 2/12, 1/12 = 1/12.
• Add: 2/12 + 1/12 = 3/12 = 1/4.

3. Elimination First

• If the question asks for time:
• Eliminate answers longer than the slowest worker’s time (e.g., if B takes 12 hours, eliminate 13+ hours).
• If the question asks for work:
• Eliminate answers larger than the sum of individual work (e.g., if A does 120 widgets and B does 150, eliminate 300+).

1-Minute Recap

"Work/rate problems test one thing: can you keep track of who’s working when? Here’s the process—every time:

1. What’s the unknown? Time? Rate? Total work? Circle it.
2. Convert all rates to the same unit. Widgets per hour? Rooms per day? Make them match.
3. Calculate individual rates. If a machine does 120 widgets in 4 hours, that’s 30/hour.
4. Combine rates if working together. Add them—don’t average or multiply.
5. Break the problem into phases. Did someone work alone first? Account for it.
6. Plug in answers if stuck. Start with B or D and work backward.

Most mistakes happen when you forget a phase or mix up rates. Slow down, write down the rates, and the answer will jump out. You’ve got this."

Final Tip:

Practice with a timer. Work/rate problems reward speed + accuracy. Aim for 90 seconds per question—if you’re over 2 minutes, you’re overcomplicating.