AMC 10 · 2009 · #1
Easy mode Grade 4Jane goes to work 5 days in a row. Each morning she buys one thing: a muffin that costs 50 cents, or a bagel that costs 75 cents. At the end of the week, the money she spent adds up to an exact number of whole dollars, with no extra cents left over. How many bagels did she buy?
Pick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: There are $5$ purchases, one per day; A muffin costs $50$ cents; a bagel costs $75$ cents; The five-day total is a whole number of dollars (no leftover cents); Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Unknowns: The number of bagels Jane bought during the week
Understand
Restated: 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.
Givens: There are $5$ purchases, one per day; A muffin costs $50$ cents; a bagel costs $75$ cents; The five-day total is a whole number of dollars (no leftover cents); Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units, #3 Eliminate Possibilities
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 $100$.
Execute — Answer: B
4.MD.A.2 Step 1 Work in cents
- Switch every price to cents so the money is whole numbers: a muffin is $50$ and a bagel is $75$.
- 'A whole number of dollars' means the total has no leftover cents — its cent value ends in $00$, so the total must be a multiple of $100$.
💡 Counting in cents turns 'whole number of dollars' into the simple test 'ends in $00$.'
4.OA.A.3 Step 2 Name the bagels, build the total
- Let $b$ be the number of bagels.
- Since there are $5$ purchases, the number of muffins is $5-b$.
- The total cost in cents is the bagels plus the muffins: $75b+50(5-b)$.
- Multiply out $50(5-b)=250-50b$ and combine: $75b+250-50b=250+25b$.
💡 One unknown, $b$, captures the whole week because the muffins are just whatever is left of the five days.
4.OA.B.4 Step 3 Set the whole-dollar condition
- For a whole number of dollars, the total $250+25b$ cents must be a multiple of $100$.
- The base $250$ already ends in $50$, and each extra bagel adds $25$ cents.
- So the question is: how many $25$-cent steps past $250$ land exactly on a multiple of $100$?
💡 A whole-dollar total is just a multiple of $100$, so we hunt for the multiple of $100$ inside the reachable range.
4.OA.B.4 Step 4 Test the six counts
- Plug in $b=0,1,2,3,4,5$: the totals are $250,275,300,325,350,375$ cents.
- The only one that is a multiple of $100$ is $300$ cents $=\$3.00$, which happens when $b=2$. Every other count leaves stray $25$, $50$, or $75$ cents. So Jane bought $2$ bagels, which is choice (B).
💡 With just six cases, checking each is faster than any clever trick and leaves no doubt.
4.MD.A.2 Switch every price to cents so the money is whole numbers: a muffin is $50$ and 4.OA.A.3 Let $b$ be the number of bagels. Since there are $5$ purchases, the number of mu 4.OA.B.4 For a whole number of dollars, the total $250+25b$ cents must be a multiple of $ 4.OA.B.4 Plug in $b=0,1,2,3,4,5$: the totals are $250,275,300,325,350,375$ cents. The onl Review
Reasonableness: The weekly total must sit between all muffins, $5\times50=250$ cents ($\$2.50$), and all bagels, $5\times75=375$ 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 $2$.
Alternative: Track only the last two digits (the cents) as bagels are added. All muffins give a total ending in $50$; each added bagel bumps the ending by $25$: $50\to75\to00$. The ending first hits $00$ after $2$ bagels, so the answer is $2$ without computing any full total.
CCSS standards used (min grade 4)
4.MD.A.2Solve word problems involving distances, time, liquid volumes, and money (Converting the $50$-cent and $75$-cent prices to cents and reading 'whole number of dollars' as a multiple of $100$ cents.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Expressing the muffins as $5-b$ and building the weekly total $75b+50(5-b)=250+25b$.)4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Requiring the total to be a multiple of $100$ and testing which bagel count $0$–$5$ satisfies it.)
⭐ Count the money in cents: a whole number of dollars just means the total ends in $00$, so find which choice lands there.
⭐ Count the money in cents: a whole number of dollars just means the total ends in $00$, so find which choice lands there.
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