AMC 8 · 2010 · #7
Grade 3 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The full range 1 to 99 is intimidating, so use Tool #9: solve the easier problem of paying every amount from 1¢ to 24¢ first — that locks in the small coins. Tool #2 (Systematic List) is the workhorse inside that easier case — list the multiples of 5 up to 20 and check which {nickel, dime} combos hit each one. Then Tool #7 (Subproblems) handles the upper amounts: once we can cover 0–24¢, adding one quarter at a time extends the range by 25¢, so 3 quarters reach all the way to 99¢. Tool #6 (Guess and Check) is the fast sanity test against the answer choices — (A) 6 is clearly too few, (C) 15 and (D) 25 are wasteful, so the smallest plausible number is around 10.
Solve an easier case first: pennies alone must cover the ones digit 0–4, so you need 4 pennies.
If you only had 3 pennies, you could never make 4¢ — pennies are the only coin worth less than a nickel.
2.MD.C.8Solve An Easier Related ProblemFor the multiples of 5 up to 20¢, 1 nickel and 2 dimes hit every one (5, 10, 15, 20); a single dime would leave 20¢ short.
Listing the multiples of 5 up to 20 and checking which {N, D} combos hit each one is a tiny Tool #2 (systematic list).
2.MD.C.8Make A Systematic ListRunning total: 4 + 1 + 2 = 7 coins, enough to build every amount from 0¢ to 24¢.
Splitting the amount into (ones-digit) + (multiple of 5) is a clean Tool #7 subproblem split.
2.NBT.B.5Identify SubproblemsEach quarter pushes the reach up by 25¢, so 3 quarters stretch the collection all the way to 99¢.
Each new quarter is a self-contained subproblem: it pushes the upper reach of the collection up by 25¢.
3.OA.D.8Identify SubproblemsAdd them all up: 4 + 1 + 2 + 3 = 10 coins, matching choice (B).
(A) 6 is too few (you already need 4 pennies and 3 quarters), and the bigger choices are clearly wasteful — (B) 10 is the only one that matches our build.
2.NBT.B.5Guess And CheckThis AMC 8 problem only needs Grade 3 reasoning: solve the easier 0–24¢ case with pennies, nickels, and dimes, then let quarters do the rest.