Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #14
Grade 6 rate-ratioPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Set up the time formula
Track units: pages × seconds-per-page cancels "pages" and leaves seconds — exactly what the question asks for.
Grade 6 unit-rate reasoning: the "page" units cancel and only seconds remain, so the setup matches the question.
6.RP.A.3Analyze The UnitsWrite each reader's time
Split into two subproblems: Bob's time = 760 × 45, Chandra's time = 760 × 30 — then subtract.
Naming each subproblem keeps the work organized and exposes the shared factor 760, which the next step will exploit.
5.NBT.B.5Identify SubproblemsSubtract the two times
Both products share 760, so factor it out: 760 × 45 - 760 × 30 = 760 × 15.
The distributive property reads a · c - b · c = (a - b) · c. Recognizing the common factor turns two multiplications into one.
The difference between Bob's total reading time and Chandra's total reading time equals 760 times the per-page gap, 760 × (45 - 30), instead of two separate large products.
▸ Why?
Both totals are built from the very same 760 groups: Bob's time is 760 groups of 45 seconds and Chandra's time is 760 groups of 30 seconds, one group per page.
▸ Why?
Total time is pages times seconds-per-page, and multiplying by 760 just means making 760 equal groups of that per-page amount.
▸ Why?
Removing 760 groups of 30 from 760 groups of 45 leaves 760 groups of what stays behind in each group, namely 45 - 30, so the shared factor 760 pulls out of the subtraction.
Finish the arithmetic
Finish: 760 × 15 = 7600 + 3800 = 11,400, which matches choice (B).
Breaking 15 into 10 + 5 is the Grade 5 "multiply by parts" strategy, which keeps the arithmetic mental-math friendly.
5.NBT.B.5Analyze The UnitsWhen 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.
- Set up the time formula
- Write each reader's time
- Subtract the two times
- Finish the arithmetic
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