Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #2
Grade 6 rate-ratio
Pick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Turn the chart into a sentence
Brenda's labeled 30% slice equals her 36 real votes, so 30% of the total 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.
The total number of votes is exactly the number whose 30% equals Brenda's 36 votes.
▸ Why?
Brenda's slice is labeled 30% of the pie and holds all 36 of her votes, so those 36 votes are 30% of the whole vote total.
▸ Why?
The pie chart sorts every single vote into exactly one of the three slices, so the three slices together are the whole total and Brenda's slice is one part of that whole.
▸ Why?
The label 30% means 30 out of every hundred of the total votes — that is exactly what "percent" says — so Brenda's slice is 30/100 of all the votes.
▸ Why?
Once 30% of the total is known to be 36, dividing 36 by 30% reverses the "take 30%" step and hands back the one total that produces it.
Find a shortcut for 30 percent
Since 10% of T is T/10, take 30% as 3 × T/10 — a quick mental check for each choice.
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.3Guess And CheckRun the check on each choice
Testing (A)–(D): 30% of 70, 84, 100, 106 gives 21, 25.2, 30, 31.8 — none equals 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.3Eliminate PossibilitiesTest the last choice left
Testing (E): 30% of 120 is 3 × 12 = 36, matching Brenda's votes, 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.3Guess And CheckConfirm the answer
Only T = 120 satisfies 30% × T = 36, so the answer is (E) 120.
Only one of the five listed totals matches the percent condition, and that's the answer we report.
6.RP.A.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 "percent of a number" you already know — just check which total makes 30% of it equal 36!
- Turn the chart into a sentence
- Find a shortcut for 30 percent
- Run the check on each choice
- Test the last choice left
- Confirm the answer
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