2009 AMC 10B Problems/Problem 1

The following problem is from both the 2009 AMC 10B #1 and 2009 AMC 12B #1, so both problems redirect to this page.

Problem

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

Solution 1 (Algebra)

A muffin costed quarters, and a bagel costed quarters.

Suppose that Jane bought muffins and bagels, where and are nonnegative integers. We need:

1. 
2. is divisible by

From Condition 2, it is clear that must be even, so we narrow down the choices to either or By a quick inspection, the only solution is from which the answer is

~MRENTHUSIASM

Solution 2 (Arithmetic)

In this solution, all amounts are in the unit of cents.

Note that the amount spent on muffins must end in either or and the amount spent on bagels must end in one of or Furthermore, the number of muffins bought and the number of bagels bought must sum to

We have two possible cases:

1. The amounts spent on muffins and bagels both end in

The number of muffins bought must be even, and the number of bagels bought must be a multiple of

In this case, there are no solutions.

2. The amounts spent on muffins and bagels both end in

The number of muffins bought must be odd, and the number of bagels bought must be more than a multiple of

In this case, the only solution is muffins and bagels, from which the answer is

~MRENTHUSIASM

Solution 3 (Observations)

If Jane bought bagel, then she bought muffins. Her total cost for the week would be cents, or dollars. So, is incorrect.

Two solutions follow from here:

Solution 3.1 (Replacements)

If Jane bought one more bagel but one fewer muffin, then her total cost for the week would increase by cents. Clearly, she bought one more bagel but one fewer muffin, or bagels in total.

~MRENTHUSIASM

Solution 3.2 (Answer Choices)

• If Jane bought bagels, then she bought muffins. Her total cost for the week would be cents, or dollars. So, is correct.

For completeness, we will check and too. If you decide to use this solution on the real test, then you will not need to do that, as you want to save more time.

• If Jane bought bagels, then she bought muffins. Her total cost for the week would be cents, or dollars. So, is incorrect.

• If Jane bought bagels, then she bought muffin. Her total cost for the week would be cents, or dollars. So, is incorrect.

• If Jane bought bagels, then she bought muffins. Her total cost for the week would be cents, or dollars. So, is incorrect.

~MRENTHUSIASM

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.