Competition · AMC preparation · step 4 of 4
AMC 8 · 2020 · #8
Grade 5 arithmeticPick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Build the largest value
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.
Multiplying 2019 × 5 is a Grade 5 multi-digit multiplication.
The greatest possible total value comes from keeping exactly one penny and making every other coin a nickel.
▸ Why?
A nickel adds more to the total than a penny, so turning any coin from a penny into a nickel always raises the total; you raise it the most by using nickels for every coin except the single penny the rules force you to keep.
▸ Why?
The total is found by adding up what each separate coin is worth, so switching one coin from a penny to a nickel lifts the total by just that coin's own gain and nothing else moves.
▸ Why?
You get the whole amount by adding every coin's value together, with no coin skipped and none counted twice.
▸ Why?
One coin going from a 1-cent penny to a 5-cent nickel is worth 5 minus 1, which is 4 more cents, since 4 is exactly what you add to a penny's value to reach a nickel's value.
▸ Why?
The 2020 coins split with no overlap into the pennies kept and the nickels made, and the rules force at least one penny, so the most coins you may turn into nickels is all but that one, 2020 minus 1.
Build the smallest value
Same idea for the minimum: keep just 1 nickel and make the other 2019 coins pennies, giving 2024 cents.
This is the second subproblem — a multi-step word problem using four operations, a Grade 4 skill.
4.OA.A.3Identify SubproblemsCheck one neighbouring split
Guess-and-check a neighbour: 2 pennies and 2018 nickels give 10092 — 4 cents less, confirming each swap shifts the total by 4.
Comparing two big totals with a quick subtraction is Grade 4 multi-digit arithmetic.
4.NBT.B.4Guess And CheckSubtract the two values
Answer the real question — subtract the min from the max: 10096 − 2024 = 8072.
Subtracting two 4- and 5-digit numbers is Grade 4 multi-digit subtraction.
4.NBT.B.4Identify SubproblemsMatch against the choices
Match 8072 to the choice list — it is option (C).
Reading and selecting the matching answer is the final step of any multi-step word problem (Grade 4).
4.OA.A.3Identify SubproblemsThis AMC 8 problem only needs Grade 5 multi-digit multiplication and a little Grade 4 word-problem thinking you already know!
- Build the largest value
- Build the smallest value
- Check one neighbouring split
- Subtract the two values
- Match against the choices
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