AMC 10 · 2017 · #2
Easy mode Grade 4Pablo wants to buy as many popsicles as he can with $8. The store sells them three ways: one popsicle for $1, a box of 3 popsicles for $2, and a box of 5 popsicles for $3. What is the greatest number of popsicles he can buy?
Pick an answer.
AMC 10 2017 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: Pablo has $\$8$ to spend on popsicles. The store sells them three ways: a single popsicle for $\$1$, a box of $3$ popsicles for $\$2$, and a box of $5$ popsicles for $\$3$. Spending up to $\$8$, find the largest total number of popsicles he can buy.
Givens: A single popsicle costs $\$1$ and gives $1$ popsicle; A box costs $\$2$ and gives $3$ popsicles; A box costs $\$3$ and gives $5$ popsicles; Pablo has $\$8$ to spend
Unknowns: The greatest number of popsicles Pablo can buy with $\$8$
Understand
Restated: Pablo has $\$8$ to spend on popsicles. The store sells them three ways: a single popsicle for $\$1$, a box of $3$ popsicles for $\$2$, and a box of $5$ popsicles for $\$3$. Spending up to $\$8$, find the largest total number of popsicles he can buy.
Givens: A single popsicle costs $\$1$ and gives $1$ popsicle; A box costs $\$2$ and gives $3$ popsicles; A box costs $\$3$ and gives $5$ popsicles; Pablo has $\$8$ to spend
Plan
Primary tool: #14 Extreme Principle
Secondary: #8 Analyze the Units, #6 Guess and Check
Tool #14 (Extreme Principle): the question asks for the greatest number of popsicles, so push toward the most popsicles per dollar and use every dollar. Tool #8 (Analyze the Units): what really matters is popsicles per dollar, so compare the three deals on equal footing before buying anything. Tool #6 (Guess and Check): after building the best buy, test a few other ways to split the $\$8$ to make sure none of them gives more.
Execute — Answer: D
4.NBT.B.5 Step 1 Compare the three deals
- To see which package is the best value, line the deals up over the same amount of money.
- Take $\$6$: with singles you get $6$ popsicles, with $\$2$ boxes you buy three boxes for $3\times3=9$ popsicles, and with $\$3$ boxes you buy two boxes for $2\times5=10$ popsicles. Over the same $\$6$ the $5$-popsicle box gives the most, so it is the best deal.
💡 Judging deals at the same price tag shows which one stretches a dollar the furthest.
4.NBT.B.4 Step 2 Buy the best box as much as possible
- Each $5$-popsicle box costs $\$3$. Two of them cost $2\times\$3=\$6$ and give $2\times5=10$ popsicles, leaving $\$8-\$6=\$2$.
- A third box would cost another $\$3$, but only $\$2$ is left, so two boxes is the most he can take.
💡 Loading up on the best deal first squeezes the most popsicles out of most of the money.
4.NBT.B.4 Step 3 Spend the last two dollars
- Only $\$2$ remains. For $\$2$ he can buy one $3$-popsicle box ($3$ popsicles) or two single popsicles ($2$ popsicles).
- The box gives more, so add it: $10+3=13$ popsicles in total, with every dollar spent.
💡 Even the leftover money should go to the better popsicles-per-dollar choice.
4.OA.A.3 Step 4 Check no split beats 13
- Test other ways to spend $\$8$. Four $3$-popsicle boxes cost $\$8$ and give $4\times3=12$ popsicles.
- One $5$-box plus two $3$-boxes costs $\$3+\$4=\$7$ and gives $5+6=11$, with $\$1$ left for $1$ more single, so $12$ again.
- None of these reaches $13$, so $13$ is the greatest, which is choice $\textbf{(D)}$.
💡 Trying the natural rivals confirms the best buy really is the biggest.
4.NBT.B.5 To see which package is the best value, line the deals up over the same amount o 4.NBT.B.4 Each $5$-popsicle box costs $\$3$. Two of them cost $2\times\$3=\$6$ and give $2 4.NBT.B.4 Only $\$2$ remains. For $\$2$ he can buy one $3$-popsicle box ($3$ popsicles) or 4.OA.A.3 Test other ways to spend $\$8$. Four $3$-popsicle boxes cost $\$8$ and give $4\t Review
Reasonableness: The answer must beat the all-singles count: $\$8$ of single popsicles is only $8$, and $13>8$, so using boxes clearly helped. It must also stay below a perfect dream rate: even at the best deal of $5$ popsicles per $\$3$, eight dollars caps out near $13$, so a total like $15$ is out of reach. Spending exactly $\$8$ for $13$ popsicles averages about $1.6$ popsicles per dollar, sitting between the $3$-box rate ($1.5$) and the $5$-box rate (about $1.67$) — exactly what mixing those two boxes should give.
Alternative: Tool #3 (Eliminate Possibilities): the biggest choice, $15$, would need $15$ popsicles from $\$8$, but even all $5$-boxes give at most $\lfloor \frac{8}{3}\rfloor\times5=10$ plus a little, so $15$ and $12$-and-below can be ruled out by spotting the reachable totals, leaving only $13$ at $\textbf{(D)}$.
CCSS standards used (min grade 4)
4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Scaling each deal to the same $\$6$ and the boxes to their popsicle counts: $3\times3=9$ and $2\times5=10$.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Tracking money and popsicles: $\$8-\$6=\$2$ and $10+3=13$.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Reasoning through buy-the-best-deal then handle-the-leftover, and checking rival ways to spend the $\$8$.)
⭐ Find the best popsicles-per-dollar deal, buy as many as you can, then spend the leftover on the next-best deal: two $\$3$ boxes plus one $\$2$ box make $13$ popsicles — choice $\textbf{(D)}$.
⭐ Find the best popsicles-per-dollar deal, buy as many as you can, then spend the leftover on the next-best deal: two $\$3$ boxes plus one $\$2$ box make $13$ popsicles — choice $\textbf{(D)}$.
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