"Two Machines, Y and Z, Work at Constant Rates, Producing Identical Items..." — 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
Two machines, Y and Z, work at constant rates, producing identical items. Machine Y produces 3 items in the same time Machine Z produces 2 items. If Machine Y takes 9 minutes to produce a batch of items, how many minutes does it take for Machine Z to produce the same number of items?
(A) 6
(B) 9
(C) 9 1/2
(D) 12
(E) 13 1/2
Try it before reading on.
Setting Up the Rate Chart
You see "constant rates." Make the chart.
Machine Y makes 3 items. Machine Z makes 2 items. They do this "in the same time." We don't know what that time is. Call it T.
| Rate | × Time | = Work | |
|---|---|---|---|
| Machine Y | ? | T minutes | 3 items |
| Machine Z | ? | T minutes | 2 items |
Solving for Each Rate
What times T equals 3? That's 3/T.
Machine Y's rate: 3/T items per minute.
What times T equals 2? That's 2/T.
Machine Z's rate: 2/T items per minute.
| Rate | × Time | = Work | |
|---|---|---|---|
| Machine Y | 3/T items per minute | T minutes | 3 items |
| Machine Z | 2/T items per minute | T minutes | 2 items |
The Second Scenario
Now the problem shifts. Machine Y takes 9 minutes to make some batch. How long does Z take to make the same batch?
The key move: add a new row for Machine Y. The rate stays the same — the problem said "constant rates." Carry it down.
| Rate | × Time | = Work | |
|---|---|---|---|
| Machine Y | 3/T items per minute | T minutes | 3 items |
| Machine Z | 2/T items per minute | T minutes | 2 items |
| Machine Y (new) | 3/T items per minute | 9 minutes | ? |
Solve for work: 3/T × 9 = 27/T items. That's the batch size.
Finding Machine Z's Time
Machine Z needs to make 27/T items. Add one more row:
| Rate | × Time | = Work | |
|---|---|---|---|
| Machine Y | 3/T items per minute | T minutes | 3 items |
| Machine Z | 2/T items per minute | T minutes | 2 items |
| Machine Y (new) | 3/T items per minute | 9 minutes | 27/T items |
| Machine Z (new) | 2/T items per minute | ? | 27/T items |
2/T × ? = 27/T
Multiply both sides by T/2:
? = 27/T × T/2 = 27/2 = 13 1/2 minutes
The T's cancel. That happens a lot on these problems.
The answer is (E).
Why This Problem Matters
Three things trip people up here:
The variable. When the problem says "in the same time" without saying what that time is, you use a variable. It feels strange the first time. It comes up often.
The second scenario. When the problem gives new information about the same machine, add a new row. The rate carries down.
The T's canceling. It can feel weird to work with a variable that disappears. But that's the chart doing its job. You don't need to know T to answer the question.
Ready for a harder one? Three pumps, working in pairs — and a shortcut that skips solving for each rate: "Pumps A, B, and C operate at their respective constant rates...".
Want the full strategy? Read: GMAT® Work/Rate Problems: Why Organization Matters
From Episode 44 of Real GMAT® Problems (The GMAT® Strategy Podcast).