"In a Class of 30 Students, 2 Did Not Borrow Any Books" — GMAT® Worked Solution
From Episode 43 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind GMAT® word problems, read: GMAT® Word Problems: Half Math, Half English, and the Setup That Prevents Translation Mistakes.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
In a class of 30 students, 2 students did not borrow any books from the library, 12 students each borrowed 1 book, 10 students each borrowed 2 books, and the rest of the students each borrowed at least 3 books. If the average (arithmetic mean) number of books borrowed per student was 2, what is the maximum number of books that any single student could have borrowed?
(A) 3
(B) 5
(C) 8
(D) 13
(E) 15
Half Math, Half English Setup
This problem has more data than short-term memory can hold comfortably. The first move is to get all of it onto the page in a structured form.
Start with what is given.
- 30 students total
- 2 students borrowed 0 books
- 12 students borrowed 1 book each
- 10 students borrowed 2 books each
- the rest borrowed at least 3 books each
- average per student = 2 books
Then the question.
- max books borrowed by a single student = ?
That is everything the problem has given us. Notice we have not started doing any computation yet. The point of this step is to capture the structure first, so the math has something clean to work on.
Rows and Columns
Turn the prose into a table. Two columns — number of students, total books borrowed by that group.
| Students | Books borrowed (total) |
|---|---|
| 2 | 0 |
| 12 | 12 |
| 10 | 20 |
| ? | ? |
| 30 | ? |
The first three rows are direct from the prompt. The fourth row covers the "rest" of the students — the ones who borrowed at least 3 books each. Question marks remind us those values are unknown.
A table like this is the highest-leverage tool we cover in word problem episodes. The data is now visible. We can see what is given, what is unknown, and what operations need to happen.
Step 1: How Many Students Are in the Last Group?
The first three rows account for 2 + 12 + 10 = 24 students.
There are 30 students in the class.
30 − 24 = 6 students in the last group.
| Students | Books borrowed (total) |
|---|---|
| 2 | 0 |
| 12 | 12 |
| 10 | 20 |
| 6 | ? |
| 30 | ? |
Step 2: Use the Average Formula to Find the Class Total
The average is 2 books per student across 30 students. Apply sum / count = average.
sum / 30 = 2
sum = 2 × 30 = 60
The class borrowed 60 books in total.
| Students | Books borrowed (total) |
|---|---|
| 2 | 0 |
| 12 | 12 |
| 10 | 20 |
| 6 | ? |
| 30 | 60 |
This is the move people miss most often on this question. The average looks like background detail. It is actually the key piece of given information — without converting it to a total, the problem has no upper bound and the question has no answer.
If we get stuck on a word problem, a useful prompt is: what given information have we not used yet? Quant problems rarely include data the question does not need. On this one, the unused information is the class average. Once we connect that to a fixed total, the rest of the steps line up.
Step 3: How Many Books Did the Last Group Borrow?
The first three rows account for 0 + 12 + 20 = 32 books.
60 − 32 = 28 books for the last group.
| Students | Books borrowed (total) |
|---|---|
| 2 | 0 |
| 12 | 12 |
| 10 | 20 |
| 6 | 28 |
| 30 | 60 |
Step 4: Maximize One Student
Six students borrowed 28 books in total. To push one student as high as possible, push the other five as low as possible.
The constraint is that every student in this group borrowed at least 3 books. So the other five students borrow 3 each — the minimum allowed.
5 × 3 = 15 books for the other five.
28 − 15 = 13 books for the maximum student.
The answer is (D) 13.
Why This Problem Matters
This question misses at roughly three times the rate of the warm-up. The reason is the number of steps and the amount of data — not the difficulty of any single step.
What ties the problem together is the workflow.
- Half math, half English to capture all six pieces of given information before computing anything.
- A two-column table to keep the structure visible across the multi-step solution.
- The average formula to convert the class average into a fixed total of 60 books.
- The optimization logic (minimize the others to maximize the target) to land the final answer.
Each of those tools showed up on easier problems earlier in the episode. The table appeared informally in Problem 2 (the beverage distributor) as a bracket-to-rate mapping. The average formula carried the snack shop warm-up. The half-math-half-English setup ran through both.
The hard problem is the same set of tools — just more of them, stacked. That is a pattern worth noticing. Most hard GMAT® word problems are not built on new concepts. They combine the same concepts from easier problems and add steps.
Building the workflow on the easier problems is what makes the harder ones doable under time pressure. If this question felt heavy, returning to the warm-up and re-running the same setup process is a useful next step. The reps compound.
Back to the strategy article: GMAT® Word Problems: Half Math, Half English, and the Setup That Prevents Translation Mistakes
Episode page: Real GMAT® Problems — Ep. 43 — Word Problems