AMC 8 · 2013 · #4

Grade 5 arithmeticalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

linear-equations-one-varfraction-arithmeticmental-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

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Problem

Eight friends split a restaurant bill into 8 equal shares. Judi forgot her money, so the other 7 friends covered Judi's share by each paying an extra $2.50 on top of their own share. Find the total bill.

Pick an answer.

(A)

$\text{\textdollar}120$

(B)

$\text{\textdollar}128$

(C)

$\text{\textdollar}140$

(D)

$\text{\textdollar}144$

(E)

$\text{\textdollar}160$

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The bill is hidden behind two layers, so Tool #7 (Identify Subproblems) splits the work cleanly. Subproblem A: the 7 extra payments of $2.50 together fill in Judi's one missing share, so we can find one person's share. Subproblem B: once we know one share, the total bill is just 8 shares. Tool #6 (Guess and Check) is a good backup: with only five answer choices, we can divide each by 8 and see which one makes Judi's share equal 7 \times$ 2.50 = $17.50.

1STEP 1

Subproblem A: the 7 friends' $2.50 extras together fill exactly one share, so Judi's share is 7 ×$ 2.50 = $17.50.

7 ×

2.50 =

17.50

2STEP 2

Every share is equal, so each person's share is also $17.50.

one share = $17.50

3STEP 3

Subproblem B: the bill is 8 equal shares, so multiply one share by 8 to get $140.

8 ×

17.50 = 8 × 17 + 8 × 0.5 = 136 + 4 =

140

4STEP 4

The total $140 matches answer choice (C).

$140 → (C)

Answer

textdollar140

Sanity check the extras. With a

140 bill split 8 ways, each share is

140 ÷ 8 =

17.50. If 7 friends absorb Judi's

17.50, each pays an extra

17.50 ÷ 7 =

2.50 — exactly what the problem says. Numbers loop back consistently, and

140 is a reasonable restaurant tab for 8 people (about

17.50 each).

💡Key takeaway

Once 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.

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