2009 AMC 10B Problem 1

Question

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?

1

2

3

4

5

With

b

bagels the total is

50(5-b)+75b

cents

A whole number of dollars means that total is a multiple of

100

If Jane buys

b

bagels, she buys

5-b

muffins, for a total of

50(5-b)+75b = 250+25b

cents. This is a whole number of dollars when

250+25b

is a multiple of

100,

that is, when

25b\equiv50\pmod{100},

or

b\equiv2\pmod4.

The only value with

0\le b\le5

is

b=2.

Thus, the correct answer is B.

Problem 1 in other years