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2011 AMC 10A Problem 2

2011 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Number Properties · difficulty 1 of 5 · about 1 min · position P1-10

A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?

A)
$11$
B)
$12$
C)
$13$
D)
$14$
E)
$15$

Answer

Key insight

Fourteen small bottles hold only 490 milliliters, short of 500, so a fifteenth bottle is required; the answer is the ceiling of 500/35.

Solution

We need the smallest whole number $k$ with $35k \ge 500$ .

Compute $35 \cdot 14 = 490$ , which is less than $500$ , so $14$ bottles do not suffice. Then $35 \cdot 15 = 525 \ge 500$ , so $15$ bottles are enough.

Equivalently, $\frac{500}{35} = 14\frac{2}{7}$ , and since a fraction of a bottle cannot be bought, round up to $15$ .

The answer is $\boxed{\textbf{(E)}\ 15}$ .

Why this works

"Enough to completely fill" means the total must reach or exceed the target, so a leftover fraction always forces one more unit: the answer is $\lceil 500/35 \rceil$ . The direction of rounding is determined by the wording, not by the size of the remainder.

The trap

Computing 500/35 = 14.28 and rounding down to 14, which leaves the large bottle 10 milliliters short of full.

Common mistakes

Computing 500/35 = 14.28 and rounding down to 14, which leaves the large bottle 10 milliliters short of full.
Arithmetic slips such as $35 \cdot 14 = 500$ ; a quick check ( $35 \cdot 14 = 350 + 140 = 490$ ) prevents this.

Techniques

Bound the quantity above/below or estimate to pin it down · Set up the equation/formula and compute; no special trick needed