Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #4
Grade 5 arithmeticalgebraPick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The bill is hidden behind two layers, so Tool #7 (Identify Subproblems) splits the work cleanly. Subproblem A: the 7 extra payments of 2.50 together fill in Judi's one missing share, so we can find one person's share. Subproblem B: once we know one share, the total bill is just 8 shares. Tool #6 (Guess and Check) is a good backup: with only five answer choices, we can divide each by 8 and see which one makes Judi's share equal 7 ×2.50 = $17.50.
Find Judi's share
Subproblem A: the 7 friends' $2.50 extras together fill exactly one share, so Judi's share is 7 × $2.50 = $17.50.
Equal-groups multiplication: 7 groups of $2.50 — a Grade 3 multiplication word-problem idea.
The seven friends' extra payments of $2.50 each cover exactly one share, so Judi's share is 7 × $2.50 = $17.50.
▸ Why?
The restaurant is still owed the entire bill, and the seven friends already paid their own seven shares, so the amount their extra payments must fill in is exactly Judi's one remaining share.
▸ Why?
The bill splits into the eight equal shares with no gaps and no overlaps, so the seven friends' own shares plus Judi's share rebuild the whole bill — leaving Judi's single share as the only part still uncovered.
▸ Why?
The seven friends split that one uncovered share as seven equal extras of $2.50, so the coverage totals 7 × $2.50, which equals $17.50.
▸ Why?
Adding seven equal amounts of $2.50 is repeated addition of the same amount, which is exactly what the product 7 × $2.50 counts.
Use one equal share
Every share is equal, so each person's share is also $17.50.
"Equal shares" means we can use Judi's share as the size of any share.
3.OA.A.3Identify SubproblemsScale up to the total bill
Subproblem B: the bill is 8 equal shares, so multiply one share by 8 to get $140.
Breaking 17.50 into 17 + 0.5 keeps the decimal multiplication friendly — Grade 5 decimals.
5.NBT.B.7Identify SubproblemsMatch against the choices
The total $140 matches answer choice (C).
Final value matches choice (C).
5.NBT.B.7Identify SubproblemsOnce you see that the 7 extra payments add up to exactly Judi's one share, the rest is just Grade 5 decimal multiplication: 8 shares of $17.50 make $140.
- Find Judi's share
- Use one equal share
- Scale up to the total bill
- Match against the choices
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