🇺🇸 AMC 8 ⇄ switch contest
2006 AMC 8

Problem 15

Problem 15 · 2006 AMC 8 Medium
Ratios, Rates & Proportions equal-time-split

Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?

Show answer
Answer: C — Page 456.
Show hints
Hint 1 of 2
"Same amount of time" is the hook. Write each reader's time as (pages they read) × (seconds per page) and set the two times equal.
Still stuck? Show hint 2 →
Hint 2 of 2
Equal time with different speeds means the faster reader covers MORE pages. The pages split in the same ratio as the speeds — here Chandra (30 s/page) to Bob (45 s/page).
Show solution
Approach: set the two reading times equal
  1. Let Chandra read x pages; Bob reads the rest, 760 − x. Chandra's time is 30x seconds, Bob's is 45(760 − x).
  2. Equal time: 30x = 45(760 − x). Expand and gather: 75x = 45 × 760, so x = 45 × 760 ÷ 75 = 456.
  3. Chandra reads through page 456.
  4. The shape to remember: since Chandra is faster (30 vs 45), she should read more pages — the page split is 45 : 30 = 3 : 2 in her favor, and 3/5 of 760 = 456. Equal-time splits always favor the faster worker, in proportion to the other's slowness.
Another way — split the pages by ratio, no equation:
  1. Equal time means pages are shared in inverse proportion to the per-page times: Chandra : Bob = 45 : 30 = 3 : 2.
  2. Chandra gets 3 parts out of 3 + 2 = 5: (3/5)(760) = 456 pages — same answer with pure ratio reasoning.
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