AMC 8 · 2020 · #15

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

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

📏 Short solution 💡 2 insights

Problem

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.

Pick an answer.

(A)

(B)

(C)

(D)

$133 \frac13$

(E)

300

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

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 $\frac{y}{x}$ 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.

1STEP 1

Pick a friendly value for x: let x = 100, so any percent of x is just that percent number.

x = 100

2STEP 2

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

15% of 100 = 15

3STEP 3

Plug into 15% of x = 20% of y: since the left side is 15, 20% of y = 15, so y = 15 ÷ 0.20 = 75.

20% · y = 15 → y =

\frac{15}{0.20}

= 75

4STEP 4

Read off the answer: with x = 100 and y = 75, y is $\frac{75}{100}$ = 75% of x.

\frac{y}{x}

\frac{75}{100}

= 75%

5STEP 5

Cross-check: 75 is choice (C); (A) 5 and (B) 35 are too small, while (D) 133 $\frac{1}{3}$ and (E) 300 wrongly make y bigger than x.

75 = (C)

Answer

Sanity check the direction. Since 15% is a smaller slice than 20%, the equation 15% · x = 20% · 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

\frac{15}{20}

\frac{3}{4}

, so (C) is right and the magnitude (a little less than the whole of x) makes sense.

💡Key takeaway

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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