AMC 8 · 2020 · #8

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

multi-digit-arithmeticmental-arithmetic identify-subproblemsbound-inequality-then-enumerate ↑ Prerequisites: multi-digit-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Ricardo has exactly 2020 coins made up of pennies (1 cent each) and nickels (5 cents each), with at least one of each kind. Find the difference, in cents, between the greatest possible total value and the least possible total value of his coin collection.

Pick an answer.

(A)

$text{806}$

(B)

$text{8068}$

(C)

$text{8072}$

(D)

$text{8076}$

(E)

$text{8082}$

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question hides three smaller jobs inside one sentence: (a) what is the largest total value possible, (b) what is the smallest total value possible, and (c) what is their difference. Tool #7 (Identify Subproblems) splits the problem into those three clean steps. To see which mix of coins makes the value largest or smallest, Tool #9 (Easier Related Problem) is handy — try just 5 coins with at-least-one-of-each, and the pattern "put all but one into the high-value coin (or low-value coin)" becomes obvious. Tool #6 (Guess and Check) lets us confirm the extreme by comparing one trial swap (e.g., 2 pennies and 2018 nickels) against the candidate extreme.

1STEP 1

Try a tiny version first: to maximize value keep just 1 penny and make the rest nickels, so 2020 coins top out at 10096 cents.

Max = 1 × 1 + 2019 × 5 = 1 + 10095 = 10096 cents

2STEP 2

Same idea for the minimum: keep just 1 nickel and make the other 2019 coins pennies, giving 2024 cents.

Min = 2019 × 1 + 1 × 5 = 2019 + 5 = 2024 cents

3STEP 3

Guess-and-check a neighbour: 2 pennies and 2018 nickels give 10092 — 4 cents less, confirming each swap shifts the total by 4.

10096 - 10092 = 4 cents per swap

4STEP 4

Answer the real question — subtract the min from the max: 10096 − 2024 = 8072.

10096 - 2024 = 8072 cents

5STEP 5

Match 8072 to the choice list — it is option (C).

Difference = 8072 → (C)

Answer

text{8072}

A neat shortcut confirms the answer. Going from the min split (2019 pennies, 1 nickel) to the max split (1 penny, 2019 nickels) flips the role of 2018 coins from penny to nickel; each flip adds 5 - 1 = 4 cents. So the difference must be 2018 × 4 = 8072 cents, matching the direct subtraction. Both the size ( ∼ 8000) and the parity (multiple of 4) make sense, and 8072 is one of the given choices, so the answer is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 5 multi-digit multiplication and a little Grade 4 word-problem thinking you already know!

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