AMC 8 · 2017 · #2

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints

percentageratio-proportiongraph-reading ratio-proportionsystematic-enumeration ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

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Problem

Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together?

$\textbf{(A) }70 \qquad \textbf{(B) }84 \qquad \textbf{(C) }100 \qquad \textbf{(D) }106 \qquad \textbf{(E) }120$

Pick an answer.

(A)

(B)

(C)

100

(D)

106

(E)

120

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Toolkit + CCSS Solution

Understand

Restated: In a student-president election, the votes for Alicia, Brenda, and Colby are shown on a pie chart as $45\%$, $30\%$, and $25\%$ respectively. Brenda's $30\%$ slice corresponds to exactly $36$ real votes. Find the total number of votes $T$ cast across all three candidates.

Givens: Alicia: $45\%$ of the votes; Brenda: $30\%$ of the votes; Colby: $25\%$ of the votes; Brenda's $30\%$ equals exactly $36$ votes; Answer choices: (A) $70$, (B) $84$, (C) $100$, (D) $106$, (E) $120$

Unknowns: The total number of votes $T$ cast in the election

Understand

Plan

Primary tool: #6 Guess and Check

Secondary: #3 Eliminate Possibilities

Because this is multiple-choice with only five candidate totals, we can take each choice $T$, compute $30\%$ of it, and keep the one that gives exactly $36$ votes. That is Tool #6 (Guess and Check) used directionally, combined with Tool #3 (Eliminate Possibilities), which is the AMC default whenever the answer is one of five listed numbers. No algebra is required: a Grade 6 student who can take $10\%$ of a number can finish this problem mentally.

Execute — Answer: E

#3 Eliminate Possibilities 6.SP.B.5 Step 1

Translate the pie chart into one short sentence.
Brenda's slice is labeled $30\%$, and that slice represents $36$ real votes.
So whatever the total $T$ is, $30\%$ of $T$ must equal $36$.

$$30\% \text{ of } T = 36$$

💡 Reading a slice of a pie chart as "this percent of the whole" is the Grade 6 data-display skill — a labeled $30\%$ slice simply means $30\%$ of the total.

#6 Guess and Check 6.RP.A.3 Step 2

Set up a fast way to take $30\%$ of any number $T$.
Since $10\%$ of $T$ is $\tfrac{T}{10}$, we get $30\% = 3 \times \tfrac{T}{10}$.
So for each answer choice, compute $\tfrac{T}{10}$, then multiply by $3$, and check whether the result is $36$.

$$30\% \text{ of } T \;=\; 3 \times \dfrac{T}{10}$$

💡 Computing a percent of a quantity is Grade 6 ratio-and-percent reasoning, and breaking $30\%$ into "three tens" makes the check mental arithmetic.

#3 Eliminate Possibilities 6.RP.A.3 Step 3

Run the check on each choice in order.
(A) $T=70$: $\tfrac{70}{10}=7$, then $3 \times 7 = 21$.
(B) $T=84$: $\tfrac{84}{10}=8.4$, then $3 \times 8.4 = 25.2$.
(C) $T=100$: $\tfrac{100}{10}=10$, then $3 \times 10 = 30$.
(D) $T=106$: $\tfrac{106}{10}=10.6$, then $3 \times 10.6 = 31.8$.
None of (A)-(D) hit $36$, so all four are eliminated.

$$30\% \text{ of } 70 = 21,\; 84 \to 25.2,\; 100 \to 30,\; 106 \to 31.8$$

💡 Plugging each multiple-choice option back into the percent condition is the textbook "eliminate possibilities" move.

#6 Guess and Check 6.RP.A.3 Step 4

Test the only remaining choice.
(E) $T=120$: $\tfrac{120}{10}=12$, then $3 \times 12 = 36$.
Brenda's $30\%$ share is exactly $36$ votes — the condition the problem gave us.
So the total is $120$.

$$30\% \text{ of } 120 \;=\; 3 \times \dfrac{120}{10} \;=\; 3 \times 12 \;=\; 36 \;\checkmark$$

💡 The guess that matches the given $36$ votes is the answer — guess-and-check on a finite list always terminates here.

#3 Eliminate Possibilities 6.RP.A.3 Step 5

Confirm the choice.
The total $T=120$ satisfies $30\% \times 120 = 36$, so the answer is $\textbf{(E)}\ 120$.

$$T = 120 \;\Rightarrow\; \textbf{(E)}$$

💡 Only one of the five listed totals matches the percent condition, and that's the answer we report.

[1] #3 6.SP.B.5 Translate the pie chart into one short sentence. Brenda's slice is labeled $30\%

[2] #6 6.RP.A.3 Set up a fast way to take $30\%$ of any number $T$. Since $10\%$ of $T$ is $\tfr

[3] #3 6.RP.A.3 Run the check on each choice in order. (A) $T=70$: $\tfrac{70}{10}=7$, then $3 \

[4] #6 6.RP.A.3 Test the only remaining choice. (E) $T=120$: $\tfrac{120}{10}=12$, then $3 \time

[5] #3 6.RP.A.3 Confirm the choice. The total $T=120$ satisfies $30\% \times 120 = 36$, so the a

Review

Reasonableness: Sanity check the magnitudes. Alicia ($45\%$) won, Brenda ($30\%$) was second, Colby ($25\%$) was third. With $T=120$, the vote splits are Alicia $54$, Brenda $36$, Colby $30$, and $54 + 36 + 30 = 120$ — every count is a whole number and they sum to the total. Brenda's $36$ votes is bigger than Colby's $30$ and smaller than Alicia's $54$, exactly matching the pie-chart ordering. The total also feels right for a school election.

Alternative: Tool #13 (Convert to Algebra) gives the same answer in one line: let $T$ be the total, then $0.30 \, T = 36$, so $T = \tfrac{36}{0.30} = 120$. Equivalently, since $30\% = \tfrac{3}{10}$, multiply both sides by $\tfrac{10}{3}$: $T = 36 \times \tfrac{10}{3} = 12 \times 10 = 120$. Same answer, but it skips the intuition-building check across all five choices.

CCSS standards used (min grade 6)

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures (Reading Brenda's labeled $30\%$ slice of the pie chart as "$30\%$ of the total votes" — the standard Grade 6 data-display interpretation.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Taking $30\%$ of each candidate total $T$ (e.g., $30\%$ of $120 = 36$) — this is the Grade 6 "percent of a quantity" reasoning.)

⭐ This AMC 8 problem only needs Grade 6 "percent of a number" you already know — just check which total makes $30\%$ of it equal $36$!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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