AMC 8 · 2020 · #2

Grade 3 arithmetic

Archetype Arithmetic Expression Decomposition

mean-median-mode-rangemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25,$ and$ 40, $respectively. They decide to split their earnings equally among themselves. In total, how much will the friend who earned$ $40$ give to the others?

$\textbf{(A) }$ 5 \qquad \textbf{(B) } $10 \qquad \textbf{(C) }$ 15 \qquad \textbf{(D) } $20 \qquad \textbf{(E) }$ 25$

Pick an answer.

(A)

(B)

$10

(C)

$15

(D)

$20

(E)

$25

View mode:

Toolkit + CCSS Solution

Understand

Restated: Four friends did yardwork and earned $\$15$, $\$20$, $\$25$, and $\$40$. They want to pool everything and end up with equal amounts. How much money does the friend who earned $\$40$ hand over to the rest of the group?

Givens: Earnings of the four friends: $\$15$, $\$20$, $\$25$, $\$40$; The four friends split their combined earnings equally (so each ends up with the same amount); Answer choices: (A) $\$5$, (B) $\$10$, (C) $\$15$, (D) $\$20$, (E) $\$25$

Unknowns: The total dollar amount the $\$40$-earner gives away to the other three friends

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

The question hides three smaller, easier questions inside it: (1) what is the total pile of money? (2) what is each friend's fair share? (3) how much more than the fair share did the $\$40$ friend start with? Tool #7 (Identify Subproblems) is perfect because solving these three pieces in order turns a one-line word problem into three single-step arithmetic problems we can already do. Tool #3 (Eliminate Possibilities) is held in reserve: once we have an answer, we can confirm it against the five multiple-choice options to make sure no arithmetic slip occurred.

Execute — Answer: C

#7 Identify Subproblems 3.NBT.A.2 Step 1

Subproblem 1 — find the total earnings.
Add the four amounts together to see how much money the group has to share.

$\$15 + \$20 + \$25 + \$40 = \$100$

💡 Adding four whole-dollar amounts that are all under $\$50$ is a Grade 3 "add within $1000$" calculation.

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

Subproblem 2 — find each friend's fair share.
"Split equally among four" means divide the total by $4$, the same operation as sharing $100$ cookies among $4$ kids.

$\$100 \div 4 = \$25 \text{ per friend}$

💡 "Share equally among four" is the Grade 3 fair-share division word-problem move.

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

Subproblem 3 — find how much the $\$40$ friend must give up. He starts with $\$40$ but should end with $\$25$, so the gap is what he hands to the others.

$\$40 - \$25 = \$15 \;\Rightarrow\; \textbf{(C)}$

💡 Subtracting $25$ from $40$ is a Grade 3 single-step subtraction within $100$.

#3 Eliminate Possibilities 3.NBT.A.2 Step 4

Verify with Tool #3 (Eliminate Possibilities).
Check the answer against the five choices: only $\$15$ matches the gap between $\$40$ and the $\$25$ fair share. (A) $\$5$ would leave him with $\$35$, (B) $\$10$ with $\$30$, (D) $\$20$ with $\$20$, (E) $\$25$ with $\$15$ — none of those equals $\$25$.

$\$40 - \$15 = \$25 \checkmark$

💡 Plugging each choice back and subtracting from $\$40$ is still Grade 3 subtraction — perfect for a quick double-check.

[1] #7 3.NBT.A.2 Subproblem 1 — find the total earnings. Add the four amounts together to see how

[2] #7 3.OA.A.3 Subproblem 2 — find each friend's fair share. "Split equally among four" means d

[3] #7 3.NBT.A.2 Subproblem 3 — find how much the $\$40$ friend must give up. He starts with $\$4

[4] #3 3.NBT.A.2 Verify with Tool #3 (Eliminate Possibilities). Check the answer against the five

Review

Reasonableness: Quick sanity check: the four amounts $\$15, \$20, \$25, \$40$ are not too different, so the fair share $\$25$ should sit roughly in the middle — which it does. The two friends below $\$25$ ($\$15$ and $\$20$) together need $\$10 + \$5 = \$15$ to reach $\$25$, and the friend at exactly $\$25$ needs nothing. So the $\$40$ friend must supply exactly $\$15$ — matching our answer (C). The books balance, which is exactly what "split equally" requires.

Alternative: Tool #6 (Guess and Check) on each answer choice. If the $\$40$ friend gives $\$15$ (choice C), he keeps $\$25$; the $\$15$ friend receives part of that and the $\$20$ friend receives the rest so everyone lands on $\$25$. None of the other choices let all four totals come out equal — for example giving $\$10$ leaves the $\$40$ friend with $\$30$, which is already above the fair share before anyone else gets paid. So (C) is the only consistent choice.

CCSS standards used (min grade 3)

3.NBT.A.2 Fluently add and subtract within 1000 (Adding the four earnings to get $\$100$, subtracting $\$25$ from $\$40$ to get the $\$15$ handed over, and checking answer choices with quick subtractions from $\$40$.)
3.OA.A.3 Solve multiplication and division word problems within 100 (Dividing $\$100$ equally among $4$ friends to get the $\$25$ fair share — a classic Grade 3 "share equally" division word problem.)

⭐ This AMC 8 problem only needs Grade 3 add-subtract and "share equally" division that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 3)

More like this