AMC 10 · 2009 · #1

Grade 4 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
modular-arithmeticmultiplesmental-arithmetic convert-to-algebra ↑ Prerequisites: multiples
📏 Medium solution 💡 2 insights
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Problem
On each of 5 workdays Jane buys exactly one item: a 50-cent muffin or a 75-cent bagel. Her total spending for the five days comes out to an exact whole number of dollars. Find how many of the five items were bagels.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Naming the number of bagels with a single letter (Tool #4) lets the muffin count and the whole total be written from that one unknown, so the whole-dollar rule becomes one clean condition to check. Working in cents (Tool #8) is the key move: 'a whole number of dollars' just means the cent total ends in 00, i.e. is a multiple of 100. With only six possible bagel counts, Tool #3 (Eliminate Possibilities) then finishes by testing which count actually lands on a multiple of 100.

1STEP 1

Work in cents

Switch prices to cents: muffin 50, bagel 75. A whole number of dollars leaves no stray cents, so the total is a multiple of 100.

muffin=50c, bagel=75c, whole dollars⇔ total is a multiple of 100
2STEP 2

Name the bagels, build the total

Let b be the number of bagels, so the muffins number 5-b. The week costs 75b+50(5-b)=250+25b cents.

75b+50(5-b)=75b+250-50b=250+25b
3STEP 3

Set the whole-dollar condition

So 250+25b must be a multiple of 100: the base 250 ends in 50, and each extra bagel pushes it up by 25 cents.

250+25b≡ 0 (mod 100)
4STEP 4

Test the six counts

b=0..5 gives 250,275,300,325,350,375 cents; only 300 ($3.00) is a multiple of 100, so Jane bought 2 bagels, choice (B).

b=0 : 250, 1 : 275, 2 : 300, 3 : 325, 4 : 350, 5 : 375 → b=2 (B)
Answer
2
The weekly total must sit between all muffins, 5×50=250 cents ($2.50), and all bagels, 5×75=375 cents ($3.75). The only whole-dollar amount in that range is $3.00. Starting from $2.50, each muffin swapped for a bagel adds 25 cents, and reaching $3.00 needs 50 more cents — exactly 2 swaps, i.e. 2 bagels. This matches the found answer of 2.
💡Key takeaway

Count the money in cents: a whole number of dollars just means the total ends in 00, so find which choice lands there.

  • Work in cents
  • Name the bagels, build the total
  • Set the whole-dollar condition
  • Test the six counts

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