AMC 10 · 2009 · #2

Grade 2 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

multiplesmental-arithmetic systematic-enumerationcasework ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 2 insights

Problem

Four coins are drawn from a piggy bank holding pennies (1c), nickels (5c), dimes (10c), and quarters (25c). Repeats are allowed. Among 15, 25, 35, 45, and 55 cents, decide which total four coins could never add up to.

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 Eliminate Possibilities

The question is 'which could NOT be,' a finite five-choice list, so the natural move is Tool #3: build every total you can and cross it off, leaving the one that resists. Four of the five fall in one line of adding. For the survivor, Tool #8 (Analyze the Units) reads the coins by their multiples of 5 to show pennies would be needed, and Tool #14 (Extreme Principle) pins the smallest total four non-penny coins can make, proving the survivor is out of reach.

1STEP 1

List the coin values

In cents a penny is 1, a nickel 5, a dime 10, a quarter 25, so four picks just means adding four of these values, repeats allowed.

penny=1, nickel=5, dime=10, quarter=25

2STEP 2

Build the other four totals

Four coins reach 25=10+5+5+5, 35=10+10+10+5, 45=25+10+5+5, 55=25+10+10+10, so cross off (B), (C), (D), (E).

25=10+5+5+5, 35=10+10+10+5, 45=25+10+5+5, 55=25+10+10+10

3STEP 3

15 needs zero pennies

Only 15 is left. Nickel, dime, quarter are all multiples of 5 and so is 15, so with just four coins the pennies used must number zero.

5, 10, 25 are multiples of 5; 15 is a multiple of 5 → #pennies=0

4STEP 4

Four nickels already beat 15

With no pennies the smallest four-coin total is four nickels, 5+5+5+5=20 cents, already above 15 — so 15 is impossible, (A).

5+5+5+5=20 ＞ 15 → 15 impossible → (A)

Answer

The four totals we built — 25, 35, 45, 55 — are all at least 20 and are all counts-by-5, matching the two rules these coins obey. Only 15 breaks a rule: being a multiple of 5 it cannot use pennies, yet four non-penny coins start at 20 cents, above 15. So 15 is the odd one out, confirming (A).

💡Key takeaway

The nickel, dime, and quarter only move in jumps of 5, so four of them start at 20 cents — and you can't drop to 15 without pennies, which would knock you off a count-by-5 total.

List the coin values
Build the other four totals
15 needs zero pennies
Four nickels already beat 15

List the coin values

Build the other four totals

15 needs zero pennies

Four nickels already beat 15

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