AMC 8 · 2013 · #4

Grade 5 arithmeticalgebra

Archetype Arithmetic Expression Decomposition

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 ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?

Pick an answer.

(A)

$\text{ extdollar}120$

(B)

$\text{ extdollar}128$

(C)

$\text{ extdollar}140$

(D)

$\text{ extdollar}144$

(E)

$\text{ extdollar}160$

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: There are $8$ friends in total, and the bill is split into $8$ equal shares; Judi pays nothing; the remaining $7$ friends cover her share; Each of those $7$ friends pays an extra $\$2.50$ beyond their own share; Answer choices: (A) $\$120$, (B) $\$128$, (C) $\$140$, (D) $\$144$, (E) $\$160$

Unknowns: The total bill in dollars

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #6 Guess and Check

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 $7 \times \$2.50 = \$17.50$.

Execute — Answer: C

#7 Identify Subproblems 3.OA.A.3 Step 1

Subproblem A — find Judi's share.
The $7$ friends each chip in $\$2.50$ extra, and those extras together must equal exactly one share (Judi's). So Judi's share is $7 \times \$2.50$.

$7 \times \$2.50 = \$17.50$

💡 Equal-groups multiplication: $7$ groups of $\$2.50$ — a Grade 3 multiplication word-problem idea.

#7 Identify Subproblems 3.OA.A.3 Step 2

Because every share is equal, Judi's share is the same as each person's share.
So one share is $\$17.50$.

$\text{one share} = \$17.50$

💡 "Equal shares" means we can use Judi's share as the size of any share.

#7 Identify Subproblems 5.NBT.B.7 Step 3

Subproblem B — scale up to the total bill.
The bill is $8$ equal shares, so multiply one share by $8$.

$8 \times \$17.50 = 8 \times 17 + 8 \times 0.5 = 136 + 4 = \$140$

💡 Breaking $17.50$ into $17 + 0.5$ keeps the decimal multiplication friendly — Grade 5 decimals.

#7 Identify Subproblems 5.NBT.B.7 Step 4

Match $\$140$ to the answer choices.

$\$140 \;\Rightarrow\; \textbf{(C)}$

💡 Final value matches choice (C).

[1] #7 3.OA.A.3 Subproblem A — find Judi's share. The $7$ friends each chip in $\$2.50$ extra, a

[2] #7 3.OA.A.3 Because every share is equal, Judi's share is the same as each person's share. S

[3] #7 5.NBT.B.7 Subproblem B — scale up to the total bill. The bill is $8$ equal shares, so mult

[4] #7 5.NBT.B.7 Match $\$140$ to the answer choices.

Review

Reasonableness: Sanity check the extras. With a $\$140$ bill split $8$ ways, each share is $\$140 \div 8 = \$17.50$. If $7$ friends absorb Judi's $\$17.50$, each pays an extra $\$17.50 \div 7 = \$2.50$ — exactly what the problem says. Numbers loop back consistently, and $\$140$ is a reasonable restaurant tab for $8$ people (about $\$17.50$ each).

Alternative: Tool #6 (Guess and Check) on the choices: divide each by $8$ to get one share, then multiply that share by $7$, and see which choice makes the result $7 \times \$2.50 = \$17.50$ short of itself by exactly one share. (A) $\$120 / 8 = \$15$, extras would be $\$15 / 7 \approx \$2.14$. (B) $\$128 / 8 = \$16$, extras $\approx \$2.29$. (C) $\$140 / 8 = \$17.50$, extras $= \$2.50$ ✓. (D) $\$144 / 8 = \$18$, extras $\approx \$2.57$. (E) $\$160 / 8 = \$20$, extras $\approx \$2.86$. Only (C) matches.

CCSS standards used (min grade 5)

3.OA.A.3 Use multiplication and division within 100 to solve word problems involving equal groups (Computing Judi's share as $7$ equal groups of $\$2.50$: $7 \times \$2.50 = \$17.50$, and recognizing all $8$ shares are equal.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying $8 \times \$17.50 = \$140$ (and the alternative-approach divisions of each choice by $8$).)

⭐ 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 $\$140$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 5)

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