AMC 8 · 2020 · #15

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

percentageratio-proportionfraction-decimal-conversion convert-to-algebra ↑ Prerequisites: percentageratio-proportion

📏 Short solution 💡 2 insights

Problem

Suppose $15\%$ of $x$ equals $20\%$ of $y.$ What percentage of $x$ is $y?$

$\textbf{(A) }5 \qquad \textbf{(B) }35 \qquad \textbf{(C) }75 \qquad \textbf{(D) }133 \frac13 \qquad \textbf{(E) }300$

Pick an answer.

(A)

(B)

(C)

(D)

133 rac13

(E)

300

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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

#9 Solve an Easier Related Problem 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.

$$x = 100$$

💡 Using a friendly $x$ turns abstract percent talk into concrete arithmetic — this is exactly the "easier related problem" move.

#9 Solve an Easier Related Problem 6.RP.A.3 Step 2

Compute the left side: $15\%$ of $x = 15\%$ of $100$.

$$15\% \text{ of } 100 = 15$$

💡 "$15\%$ of $100$" means $15$ out of every $100$, so the answer is just $15$.

#9 Solve an Easier Related Problem 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 $y = 15 \div 0.20 = 75$.

$$20\% \cdot y = 15 \;\Rightarrow\; y = \dfrac{15}{0.20} = 75$$

💡 If one-fifth of $y$ is $15$, then $y$ itself is five $15$s, which is $75$.

#9 Solve an Easier Related Problem 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 $x$.

$$\dfrac{y}{x} = \dfrac{75}{100} = 75\%$$

💡 Choosing $x = 100$ makes "$y$ out of $x$" the same as "$y$ percent" — no extra conversion needed.

#3 Eliminate Possibilities 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 $y$.

$$75 = \textbf{(C)}$$

💡 On AMC multiple choice, locating your answer in the list and ruling out the others is the standard final check.

[1] #9 6.RP.A.3 Pick the easiest value for $x$. Choose $x = 100$ so that taking percentages of $

[2] #9 6.RP.A.3 Compute the left side: $15\%$ of $x = 15\%$ of $100$.

[3] #9 6.RP.A.3 Use the given equation $15\% \text{ of } x = 20\% \text{ of } y$ to find $y$. Th

[4] #9 6.RP.A.3 Read off the requested percentage. We chose $x = 100$ and found $y = 75$, so $y$

[5] #3 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.3 Use 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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