AMC 10 · 2011 · #2

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

multi-digit-arithmeticestimationmultiples bound-inequality-then-enumerate ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 1 insight

Problem

A small shampoo bottle holds 35 ml and a large bottle holds 500 ml. Find the fewest small bottles Jasmine must buy so that their combined shampoo completely fills the large bottle.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Extreme Principle

The question asks for the MINIMUM number of bottles, and the whole trap lives at the boundary: dividing 500 by 35 gives about 14.3, which sits between two answer choices. Tool #14 (Extreme Principle) is exactly the min/max lens — we look for the smallest whole number of bottles that first pushes the total to 500 ml or more. Tool #8 (Analyze the Units) keeps the bookkeeping honest: each bottle adds 35 ml, so n bottles give 35n ml, and we compare that to 500 ml. Tool #6 (Guess and Check) then confirms the boundary by testing the two neighboring counts, 14 and 15.

1STEP 1

Write the filling condition

Each small bottle pours in 35 ml, so n bottles give 35 × n ml — filling the large bottle needs that total to reach 500 ml.

35 × n ≥ 500

2STEP 2

Divide 500 by 35

Divide: 500 = 35 × 14 + 10, so 14 whole bottles fit (490 ml) and 10 ml is still left uncovered.

500 ÷ 35 = 14 remainder 10, since 500 = 35 × 14 + 10

3STEP 3

Read the remainder

The leftover matters: 14 bottles give only 490 ml, 10 ml short of full, so the count rounds up to 15.

14 bottles → 490 ml ＜ 500 ml, so round up to 15 bottles

4STEP 4

Test both boundary counts

Check the boundary: 14 × 35 = 490 ＜ 500 falls short, while 15 × 35 = 525 ≥ 500 fills it, so the minimum is 15 bottles.

14 × 35 = 490 ＜ 500, 15 × 35 = 525 ≥ 500 → (E) 15

Answer

Estimate: 500 ÷ 35 ≈ 14.3, so the answer must be just above 14. The choices (D) 14 and (E) 15 are the two candidates, and 14 is the trap for anyone who rounds down. Since 14 bottles give 490 ml — 10 ml short — the bottle would not be full, so 15 is the smallest count that works. That matches (E).

💡Key takeaway

When a real-world division has a leftover, round UP — 14 bottles fall 10 ml short, so you need a 15th to truly fill the big bottle.

Write the filling condition
Divide 500 by 35
Read the remainder
Test both boundary counts

Write the filling condition

Divide 500 by 35

Read the remainder

Test both boundary counts

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