AMC 8 · 2006 · #14

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

ratemulti-digit-arithmeticformula-substitution identify-subproblems ↑ Prerequisites: ratemulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

A 760-page novel is read by both Bob and Chandra. Bob takes 45 seconds per page; Chandra takes 30 seconds per page. Find how many more total seconds Bob spends reading the whole book than Chandra does.

Pick an answer.

(A)

7,600

(B)

11,400

(C)

12,500

(D)

15,200

(E)

22,800

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a rate problem, so Tool #8 (Analyze the Units) is the natural anchor: pages × (seconds / page) cancels to seconds, which tells us how to combine 760 pages with each reader's per-page rate. Tool #7 (Identify Subproblems) splits the main question into two clean pieces — Bob's total time and Chandra's total time — that we then subtract. The factoring shortcut 760 · 45 - 760 · 30 = 760(45 - 30) is just the distributive property applied after the subproblems are written down, so no algebra tool is needed.

1STEP 1

Track units: pages × seconds-per-page cancels "pages" and leaves seconds — exactly what the question asks for.

total seconds = pages × seconds/page

2STEP 2

Split into two subproblems: Bob's time = 760 × 45, Chandra's time = 760 × 30 — then subtract.

Bob's time = 760 × 45, Chandra's time = 760 × 30

3STEP 3

Both products share 760, so factor it out: 760 × 45 - 760 × 30 = 760 × 15.

760 × 45 - 760 × 30 = 760 × (45 - 30) = 760 × 15

4STEP 4

Finish: 760 × 15 = 7600 + 3800 = 11,400, which matches choice (B).

760 × 15 = 7600 + 3800 = 11,400 → (B)

Answer

11,400

Sanity check the magnitude. Bob is slower by 15 seconds on every page, and there are 760 pages, so the gap should be roughly 760 × 15 ≈ 11,000 seconds. The exact value 11,400 lines up. Bob's full time is 760 × 45 = 34,200 seconds and Chandra's is 760 × 30 = 22,800 seconds; their difference is 34,200 - 22,800 = 11,400, which agrees with the factored computation. The units are seconds, matching what the problem asks for.

💡Key takeaway

When two products share a factor like 760, pull it out first: 760 × 45 - 760 × 30 = 760 × 15. The hard-looking subtraction becomes one easy multiplication.

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