AMC 8 · 2017 · #2
Grade 6 rate-ratioProblem
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?
Pick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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) $120$
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
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 $36$.
💡 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.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 $36$.
💡 Computing a percent of a quantity is Grade 6 ratio-and-percent reasoning, and breaking $30\%$ into "three tens" makes the check mental arithmetic.
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.
💡 Plugging each multiple-choice option back into the percent condition is the textbook "eliminate possibilities" move.
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 $120$.
💡 The guess that matches the given $36$ votes is the answer — guess-and-check on a finite list always terminates here.
6.RP.A.3 Step 5 - Confirm the choice.
- The total $T=120$ satisfies $30\% \times 120 = 36$, so the answer is $\textbf{(E)}\ 120$.
💡 Only one of the five listed totals matches the percent condition, and that's the answer we report.
6.SP.B.5 Translate the pie chart into one short sentence. Brenda's slice is labeled $30\% 6.RP.A.3 Set up a fast way to take $30\%$ of any number $T$. Since $10\%$ of $T$ is $\tfr 6.RP.A.3 Run the check on each choice in order. (A) $T=70$: $\tfrac{70}{10}=7$, then $3 \ 6.RP.A.3 Test the only remaining choice. (E) $T=120$: $\tfrac{120}{10}=12$, then $3 \time 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.5Summarize 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.3Use 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 $36$!
⭐ This AMC 8 problem only needs Grade 6 "percent of a number" you already know — just check which total makes $30\%$ of it equal $36$!
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