AMC 8 · 2006 · #15

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

ratelinear-equations-one-varratio-proportion convert-to-algebraidentify-subproblems ↑ Prerequisites: ratelinear-equations-one-var

📏 Medium solution 💡 3 insights

Problem

Chandra and Bob split a 760-page novel so they spend equal reading time. Chandra reads pages 1 through some page x at 30 seconds per page; Bob reads pages x+1 through 760 at 45 seconds per page. Find the last page x that Chandra reads.

Pick an answer.

(A)

425

(B)

444

(C)

456

(D)

484

(E)

506

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

How to solve

Strategy Analyze the Units

The rates are in seconds per page, and the condition is equal total time in seconds. Tool #8 (Analyze the Units) makes the setup almost automatic: (sec/page) × (pages) = sec, so equal time means 30 × (Chandra's pages) = 45 × (Bob's pages). That immediately exposes a 3:2 page-count ratio without heavy algebra. Tool #13 (Convert to Algebra) gives a single name x to Chandra's page count so we can write the equation, but the work that finishes the problem is the ratio, not symbol manipulation.

1STEP 1

Let x be Chandra's page count; Bob reads the rest, 760 - x pages.

Chandra: x pages, Bob: 760 - x pages

2STEP 2

Equal time means rate × pages match on both sides: 30x = 45(760 - x).

30 sec/pg · x pg = 45 sec/pg · (760 - x) pg

3STEP 3

Chandra is faster, so page counts flip to the inverse of the speeds — the ratio 3:2.

\frac{x}{760 - x}

\frac{45}{30}

\frac{3}{2}

4STEP 4

Split 760 into 5 shares of 152; Chandra's 3 shares give 456.

x =

\frac{3}{5}

× 760 = 3 × 152 = 456 → (C)

Answer

456

Check the time: Chandra reads 456 pages at 30 sec/page for 456 × 30 = 13,680 seconds. Bob reads 760 - 456 = 304 pages at 45 sec/page for 304 × 45 = 13,680 seconds. The two totals match, so the page split is correct. The answer also passes a sanity check: Chandra is 1.5× as fast as Bob (

\frac{45}{30}

= 1.5), and indeed

\frac{456}{304}

= 1.5 — she read 1.5 times as many pages, just as the equal-time condition requires.

💡Key takeaway

When two readers spend equal time but read at different speeds, the faster reader handles more pages — and the seconds-per-page numbers tell you exactly how to split the total in a Grade 6 ratio.

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