2009 AMC 12B Problem 1

Problem 1 of 25EasierAlgebra Number Theory

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?

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Solution

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.