AMC 10 · 2009 · #1

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

unit-conversionmultiplesestimation guess-and-check ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 1 insight

Problem

One can holds 12 ounces of soda. A gallon is 128 ounces. Find the smallest number of whole cans whose soda adds up to at least 128 ounces.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

How to solve

Strategy Analyze the Units

The problem links two units: ounces per can and total ounces needed. Track how ounces build up as cans are added, test whole numbers of cans against the 128-ounce target, and take the smallest count that reaches it.

1STEP 1

Set the target

Each can adds 12 ounces, so n cans hold 12 times n — and that total must reach at least 128 ounces.

12 × n ≥ 128

2STEP 2

Test 10 cans

Try 10 cans: 12 times 10 is 120 ounces, less than 128, so ten cans fall short of a gallon.

12 × 10 = 120 ＜ 128

3STEP 3

Test 11 cans

Add one more can: 12 times 11 is 132 ounces, which clears 128, so eleven cans make a gallon.

12 × 11 = 132 ≥ 128

4STEP 4

Pick the smallest that works

Ten falls short and eleven clears the bar, so the fewest cans reaching a gallon is 11 — choice (E).

min{ n : 12n ≥ 128 } = 11

Answer

A gallon of 128 ounces is a little more than ten 12-ounce cans, which give 120 ounces, so the count should be just above 10, and 11 fits. Eleven cans give 132 ounces, only 4 ounces of spare soda, matching a 'just barely enough' answer. The smaller choices 7, 8, 9, and 10 all give fewer than 128 ounces, so none of them can make a gallon.

💡Key takeaway

When you can only buy whole items, work out how many you need and round UP so you have enough.

Set the target
Test 10 cans
Test 11 cans
Pick the smallest that works

Set the target

Test 10 cans

Test 11 cans

Pick the smallest that works

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