2009 AMC 12B Problem 1

Attempt Problem 1 of the 2009 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

She buys

5

items total, so muffins and bagels together number

5

The total

50m + 75b

cents must be a multiple of

100;

reduce the cost modulo

100

With

b

bagels she buys

5 - b

muffins, costing

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

cents. For a whole number of dollars this must be a multiple of

100.

Thus

25b\equiv50\pmod{100},

b\equiv2\pmod4.

Since

0\le b\le5,

only

b=2

works, giving

300

cents

= \$3.

Thus, the correct answer is B.

Problem 1 in other years