AMC 10 · 2017 · #2

Grade 4 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

ratio-proportionoptimization-counting greedy-algorithm ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 1 insight

Problem

Pablo has

8 to spend</span> on popsicles. The store sells them three ways: <span class="mk-c">a single popsicle for

1, a box of 3 popsicles for

2</span>, and <span class="mk-c">a box of 5 popsicles for

3. Spending up to $8, find the largest total number of popsicles he can buy.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Extreme Principle

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.

1STEP 1

Compare the three deals

Line the deals up over $6: singles give 6, three$ 2 boxes give 3×3=9, two $3 boxes give 2×5=10 — the 5-popsicle box wins.

$6: 6 vs 3×3=9 vs 2×5=10

2STEP 2

Buy the best box as much as possible

Each $3 box gives 5, so two cost$ 6 for 10 popsicles, leaving $8-$ 6= $2 — a third box needs$ 3, so two is the max.

2×

3=

6 → 10 popsicles,

8-

6=$2

3STEP 3

Spend the last two dollars

With $2 left, a 3-popsicle box beats two singles, so add it: 10+3=13 popsicles, every dollar spent.

$2: 3 vs 2 → 10+3=13

4STEP 4

Check no split beats 13

Rivals fall short: four $2 boxes give 4×3=12, and a$ 3 box plus two $2 boxes plus a single also give 12 — 13 stays best, choice (D).

4×3=12, 5+6+1=12, best=13 → (D)

Answer

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.

💡Key takeaway

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 (D).

Compare the three deals
Buy the best box as much as possible
Spend the last two dollars
Check no split beats 13

Compare the three deals

Buy the best box as much as possible

Spend the last two dollars

Check no split beats 13

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