2011 AMC 10A Problem 2

P1-10~1 minPrint

Number Properties Linear Equations & Word Problems Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Fourteen small bottles hold only 490 milliliters, which is not quite 500, so she needs one more bottle.

Solution

We want the fewest small bottles that together hold at least $500$ milliliters.

Estimate first: $35 \times 10 = 350$ , which is too little, so the answer is more than $10$ .
Try $14$ bottles: $35 \times 14 = 490$ milliliters. That is $10$ milliliters short of $500$ , so $14$ is not enough.
Try $15$ bottles: $35 \times 15 = 525$ milliliters. That is at least $500$ , so $15$ bottles fill the large bottle (with some left over).

So the smallest number that works is $15$ .

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

Why this works

"Completely fill" means the total has to reach the target. If $500 \div 35$ is not a whole number, the leftover part still needs a whole extra bottle, so you always round up. Whether you round up or down comes from the wording of the question, not from how big the remainder is.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(E)}\ 15}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 20: Special Functions (§20.3)
Introduction to Algebra, Chapter 7: Proportion (§7.4)