"Pumps A, B, and C Operate at Their Respective Constant Rates..." — GMAT® Worked Solution
From Episode 44 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the strategy behind the rate chart, read: GMAT® Work/Rate Problems: Why Organization Matters.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours. Pumps A and C, operating simultaneously, can fill the tank in 3/2 hours. Pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?
(A) 1/3
(B) 1/2
(C) 2/3
(D) 5/6
(E) 1
Try it before reading on.
Setting Up the Rate Chart
You see "constant rates." Make the chart.
Each row is a pair of pumps. Fill in the time and work:
| Rate | × Time | = Work | |
|---|---|---|---|
| A + B | ? | 6/5 hours | 1 tank |
| A + C | ? | 3/2 hours | 1 tank |
| B + C | ? | 2 hours | 1 tank |
| A + B + C | ? | ? | 1 tank |
Write the last row now. That's what you're solving for.
Solving for Each Pair's Rate
Same move as before. What times the time equals 1?
- A + B: what × 6/5 = 1? 5/6 tanks per hour.
- A + C: what × 3/2 = 1? 2/3 tanks per hour.
- B + C: what × 2 = 1? 1/2 tanks per hour.
| Rate | × Time | = Work | |
|---|---|---|---|
| A + B | 5/6 tanks per hour | 6/5 hours | 1 tank |
| A + C | 2/3 tanks per hour | 3/2 hours | 1 tank |
| B + C | 1/2 tanks per hour | 2 hours | 1 tank |
| A + B + C | ? | ? | 1 tank |
The Long Way
You could set up three equations and solve for A, B, and C one at a time. That works. But it's a lot of steps, and each step is a chance for a fraction error.
The Shortcut
Add all three pair rates together:
(A + B) + (A + C) + (B + C) = 2A + 2B + 2C
Each pump shows up twice. Divide by 2:
A + B + C
Now add the right side:
5/6 + 2/3 + 1/2
Common denominator of 6:
5/6 + 4/6 + 3/6 = 12/6 = 2
So 2A + 2B + 2C = 2.
A + B + C = 1 tank per hour.
Finding the Time
| Rate | × Time | = Work | |
|---|---|---|---|
| A + B | 5/6 tanks per hour | 6/5 hours | 1 tank |
| A + C | 2/3 tanks per hour | 3/2 hours | 1 tank |
| B + C | 1/2 tanks per hour | 2 hours | 1 tank |
| A + B + C | 1 tank per hour | 1 hour | 1 tank |
1 tank per hour × ? = 1 tank. That's 1 hour.
The answer is (E).
Why This Problem Matters
70% miss this. Here's where they get stuck:
Too many moving parts. Three pairs and one unknown combination. The chart breaks it down — each pair is just another row.
Going the long way. Solving for A, B, and C one at a time takes many more steps. The shortcut — add all three and divide by 2 — gets you there with one fraction addition.
Fraction errors. Even people who see the approach stumble on 5/6 + 2/3 + 1/2. If fractions don't feel automatic, spend time on the Math Basics episodes on The GMAT® Strategy Podcast. It'll help across many question types.
This shortcut isn't something most people learn in school. It's a GMAT®-specific pattern. The more problems you practice, the more you recognize when the setup allows for it.
Want the full strategy? Read: GMAT® Work/Rate Problems: Why Organization Matters
From Episode 44 of Real GMAT® Problems (The GMAT® Strategy Podcast).