AMC 10 · 2010 · #4

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

estimationdecimal-arithmetic extreme-principle ↑ Prerequisites: estimation

📏 Short solution 💡 2 insights

Problem

A book takes 412 minutes to read aloud, and it must be split across compact discs where each disc holds at most 56 minutes. We use as few discs as possible, and every disc must carry the exact same amount of reading. We want the number of minutes of reading on each disc.

Pick an answer.

(A)

$\ 50.2$

(B)

$\ 51.5$

(C)

$\ 52.4$

(D)

$\ 53.8$

(E)

$\ 55.2$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Extreme Principle

The phrase "smallest possible number of discs" is a boundary question, so Tool #14 (Extreme Principle) is the lead: find the fewest whole discs whose capacity can still cover 412 minutes. Because 412 ÷ 56 is not a whole number, we round up to the next whole disc. Tool #8 (Analyze the Units) keeps the bookkeeping honest — we are tracking minutes per disc, so once the disc count is fixed we simply share the 412 minutes equally. Tool #3 (Eliminate Possibilities) is a fast cross-check: only one answer choice can equal 412 split into a whole number of equal parts.

1STEP 1

Find the fewest discs

412 ÷ 56 = 7.357…, and 7 discs hold only 392 minutes, so round the count up to 8 discs.

412 ÷ 56 = 7.357… → round up to 8 discs

2STEP 2

Share the time evenly

Every disc carries the same amount, so split 412 minutes into 8 equal parts: 412 ÷ 8 = 51.5 minutes per disc.

412 ÷ 8 = 51.5

3STEP 3

Check and read off the answer

Check it fits: 51.5 ≤ 56 stays under the limit and 51.5 × 8 = 412 rebuilds the book, so the answer is (B).

51.5 ≤ 56 and 51.5 × 8 = 412 → (B)

Answer

51.5

The per-disc time must be at most 56 and, since 8 discs is the fewest possible, the load per disc should be close to the 56 limit rather than far below it. 51.5 minutes is just under 56, and using one fewer disc would demand 412 ÷ 7 ≈ 58.9 minutes per disc, which overflows the 56 cap — confirming 8 really is the minimum. Multiplying back, 8 × 51.5 = 412, so no reading is lost. Everything checks out with (B).

💡Key takeaway

You cannot buy part of a disc, so round the number of discs up to 8, then share the 412 minutes equally: 412 ÷ 8 = 51.5 minutes on each disc.

Find the fewest discs
Share the time evenly
Check and read off the answer

Find the fewest discs

Share the time evenly

Check and read off the answer

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