2009 AMC 10B Problem 1

P1-10~1 minPrint

Linear Equations & Word Problems Divisibility & Factors Use the answer choices (mod checks, size, form) to eliminate or select Test small/specific values or special cases to find or verify the answer

Each morning of her five-day workweek, Jane bought either a $50$ -cent muffin or a $75$ -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

Show solution without answering

Key insight

Five muffins cost 2 dollars and 50 cents, and each swap to a bagel adds 25 cents, so two swaps reach a whole dollar amount.

Solution

There are only five possible answers, so start from the cheapest week and see how the total changes.

If Jane bought no bagels, she bought $5$ muffins for $5 \times 50 = 250$ cents, which is $\$2.50$ . That is not a whole number of dollars.
Each time one muffin is replaced by a bagel, the total goes up by $75 - 50 = 25$ cents.
Make a table of the totals:

Bagels	$0$	$1$	$2$	$3$	$4$	$5$
Total (cents)	$250$	$275$	$300$	$325$	$350$	$375$

A whole number of dollars means the total in cents is a multiple of $100$ . Only $300$ cents works, and that happens with $2$ bagels.
Check: $2$ bagels cost $\$1.50$ and $3$ muffins cost $\$1.50$ , for $\$3.00$ in all.

The answer is $\boxed{\textbf{(B)}\ 2}$ .

Why this works

When there are two kinds of items and a fixed total number of them, start with all of one kind and count how much each swap changes the total. Here every swap adds the same $25$ cents, so the totals go up in equal steps and you can spot the one that lands on a whole dollar.

Concise solution

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

Suppose Jane bought $b$ bagels and $5 - b$ muffins. In cents her total is
$75b + 50(5 - b) = 250 + 25b.$
This must be a multiple of $100$ . Since $250$ already leaves a remainder of $50$ , the term $25b$ must contribute another $50$ (or $150$ , $250$ , and so on). With $0 \le b \le 5$ the only option is $25b = 50$ , so $b = 2$ .

Check: two bagels and three muffins cost $1.50 + 1.50 = \$3.00$ .

The answer is $\boxed{\textbf{(B)}\ 2}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)