AMC 8 · 2020 · #15
Grade 6 rate-ratioProblem
Suppose 15% of x equals 20% of y. What percentage of x is y?
Pick an answer.
AMC 8 2020 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: Two numbers $x$ and $y$ are linked by the fact that $15\%$ of $x$ gives exactly the same amount as $20\%$ of $y$. Using that link, figure out what percent of $x$ the number $y$ is.
Givens: $15\%$ of $x$ equals $20\%$ of $y$; Answer choices: (A) $5$, (B) $35$, (C) $75$, (D) $133\tfrac{1}{3}$, (E) $300$ (all percentages)
Unknowns: The percentage $p$ such that $y = p\%$ of $x$ (i.e. the value of $\tfrac{y}{x} \times 100$)
Understand
Restated: Two numbers $x$ and $y$ are linked by the fact that $15\%$ of $x$ gives exactly the same amount as $20\%$ of $y$. Using that link, figure out what percent of $x$ the number $y$ is.
Givens: $15\%$ of $x$ equals $20\%$ of $y$; Answer choices: (A) $5$, (B) $35$, (C) $75$, (D) $133\tfrac{1}{3}$, (E) $300$ (all percentages)
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #3 Eliminate Possibilities
The problem talks about $x$ in the abstract, but "$15\%$ of $x$ equals $20\%$ of $y$" is true for every choice of $x$ — the ratio $\tfrac{y}{x}$ is the same in every case. Tool #9 (Easier Related Problem) lets us replace the abstract $x$ with a friendly number like $x = 100$, because $15\%$ of $100$ is just $15$ (no decimals, no algebra). Once we have a concrete $y$, asking "what percent of $x$ is $y$?" becomes "$y$ out of $100$" — the answer is read straight off. Tool #3 (Eliminate Possibilities) is the natural cross-check on a multiple-choice question: we confirm our value sits in the answer list, and we rule out the obvious traps.
Execute — Answer: C
6.RP.A.3 Step 1 - Pick the easiest value for $x$.
- Choose $x = 100$ so that taking percentages of $x$ is just reading the percent number itself.
💡 Using a friendly $x$ turns abstract percent talk into concrete arithmetic — this is exactly the "easier related problem" move.
6.RP.A.3 Step 2 Compute the left side: $15\%$ of $x = 15\%$ of $100$.
💡 "$15\%$ of $100$" means $15$ out of every $100$, so the answer is just $15$.
6.RP.A.3 Step 3 - Use the given equation $15\% \text{ of } x = 20\% \text{ of } y$ to find $y$.
- The left side equals $15$, so $20\%$ of $y$ must equal $15$.
- Asking "$20\%$ of what is $15$?" gives $y = 15 \div 0.20 = 75$.
💡 If one-fifth of $y$ is $15$, then $y$ itself is five $15$s, which is $75$.
6.RP.A.3 Step 4 - Read off the requested percentage.
- We chose $x = 100$ and found $y = 75$, so $y$ is $\tfrac{75}{100} = 75\%$ of $x$.
💡 Choosing $x = 100$ makes "$y$ out of $x$" the same as "$y$ percent" — no extra conversion needed.
6.RP.A.3 Step 5 - Cross-check against the answer list.
- $75$ is exactly choice (C).
- Choices (A) $5$ and (B) $35$ are too small (they would force $y < x$ by a huge margin), and choices (D) $133\tfrac{1}{3}$ and (E) $300$ would force $y > x$, contradicting that $15\%$ of $x$ equals the larger percent $20\%$ of $y$.
💡 On AMC multiple choice, locating your answer in the list and ruling out the others is the standard final check.
6.RP.A.3 Pick the easiest value for $x$. Choose $x = 100$ so that taking percentages of $ 6.RP.A.3 Compute the left side: $15\%$ of $x = 15\%$ of $100$. 6.RP.A.3 Use the given equation $15\% \text{ of } x = 20\% \text{ of } y$ to find $y$. Th 6.RP.A.3 Read off the requested percentage. We chose $x = 100$ and found $y = 75$, so $y$ 6.RP.A.3 Cross-check against the answer list. $75$ is exactly choice (C). Choices (A) $5$ Review
Reasonableness: Sanity check the direction. Since $15\%$ is a smaller slice than $20\%$, the equation $15\% \cdot x = 20\% \cdot y$ forces $x$ to be the bigger number and $y$ to be the smaller. So $y$ should be less than $100\%$ of $x$, which kills (D) and (E). Among (A), (B), (C), only $75\%$ comes from the clean ratio $\tfrac{15}{20} = \tfrac{3}{4}$, so (C) is right and the magnitude (a little less than the whole of $x$) makes sense.
Alternative: Tool #5 (Look for a Pattern) using the ratio shortcut: rewrite "$15\%$ of $x = 20\%$ of $y$" as $\tfrac{15}{100} \cdot x = \tfrac{20}{100} \cdot y$, which immediately gives $\tfrac{y}{x} = \tfrac{15}{20} = \tfrac{3}{4} = 75\%$. Same answer, same Grade 6 ratio reasoning, just without picking a sample $x$.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (All of the percent-of-a-number reasoning in this problem: computing $15\%$ of $100$, solving "$20\%$ of $y = 15$" for $y$, and reading $\tfrac{75}{100}$ as $75\%$ — these are exactly the percent applications listed under this Grade 6 ratio standard.)
⭐ This AMC 8 problem only needs Grade 6 percent-of-a-number reasoning you already know — pick $x = 100$ and the rest is plain arithmetic!
⭐ This AMC 8 problem only needs Grade 6 percent-of-a-number reasoning you already know — pick $x = 100$ and the rest is plain arithmetic!
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