"Five Pieces of Wood Have an Average Length of 124 Centimeters" — GMAT® Worked Solution
From Episode 42 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind GMAT® statistics questions, read: GMAT® Statistics: The Average, the Median, and the Shortcut That Saves You Time.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
Five pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length, in centimeters, of the shortest piece of wood?
(A) 90
(B) 100
(C) 110
(D) 130
(E) 140
Setting Up the Constraints
Write down what is given.
- 5 pieces of wood
- Average length = 124 cm
- Median length = 140 cm
- Find: the maximum possible length of the shortest piece
Visualize the five pieces in order from shortest to longest:
| Position | 1st | 2nd | 3rd (median) | 4th | 5th |
|---|---|---|---|---|---|
| Length | ? | ? | 140 | ? | ? |
The 3rd piece — the middle one in an ordered list of 5 — is the median. So the 3rd value is 140.
Step 1: Find the Total
The average formula tells us:
sum / 5 = 124
Multiply both sides by 5:
sum = 124 × 5 = 620
The total length of all five pieces is 620 cm.
Step 2: Minimize the Upper Two Pieces
We want to push the shortest piece as high as it can go. With a fixed total of 620, the only way to do that is to keep as little length as possible in the other four pieces.
The 3rd piece is locked at 140 (the median).
The 4th and 5th pieces have to be at least as large as 140 — otherwise the median would no longer be 140. The smallest they can be is 140 each. The problem does not say the pieces have to be different, so values are allowed to repeat.
| Position | 1st | 2nd | 3rd | 4th | 5th |
|---|---|---|---|---|---|
| Length | ? (max) | ? (min) | 140 | 140 | 140 |
This step trips up a lot of people. The instinct is to write 141 and 142 for the upper two pieces. The problem does not require that. Unless the wording says "distinct" or "different," equal values are fine.
Step 3: Minimize the 2nd Piece
We also want the 2nd piece to be as small as possible — but it has to be at least as large as the 1st piece. So the smallest the 2nd piece can be is equal to the 1st piece, which is the value we are solving for.
Let x = the length of the shortest piece. Then the 2nd piece can also be x.
| Position | 1st | 2nd | 3rd | 4th | 5th |
|---|---|---|---|---|---|
| Length | x | x | 140 | 140 | 140 |
Step 4: Solve for x
Sum the lengths and set equal to 620:
x + x + 140 + 140 + 140 = 620
2x + 420 = 620
2x = 200
x = 100
The maximum possible length of the shortest piece is 100 cm.
The answer is (B).
Why This Problem Matters
About 10 percent of test takers select (A) 90 on this one. The most common cause is one of two setup issues.
The first is misreading the question. People sometimes solve for the shortest piece without optimizing — or accidentally maximize the longest piece instead. Boxing the question text on your scratch paper helps prevent that.
The second is assuming the upper two pieces have to be larger than 140. That assumption raises the total committed to the upper end of the list, which forces the lower end smaller and pushes you toward (A) instead of (B).
The fix on that second one is a habit: unless the problem uses the word "different" or "distinct," default to allowing equal values. On constraint problems, that flexibility is usually where the optimal setup lives.
A note for the next problem: this same pattern — find the total with the average formula, then minimize the other values to maximize your target — scales directly to the hardest statistics question we cover in this episode. The structure is the same. The numbers just get harder.
Next problem: 30 Students Borrowed Books From the Library — Maximize One Student — GMAT® Worked Solution
Back to the strategy article: GMAT® Statistics: The Average, the Median, and the Shortcut That Saves You Time
Episode page: Real GMAT® Problems — Ep. 42 — Statistics