AMC 8 · 2018 · #1

Grade 6 rate-ratio

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

ratio-proportionfraction-arithmeticestimation dimensional-analysisidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Short solution 💡 2 insights

Problem

An amusement park makes scale models at a 1:20 ratio, meaning every model is

\frac{1}{20}

the size of the real building. The real U.S. Capitol is 289 feet tall. How tall, to the nearest whole foot, is its scale model?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The phrase "ratio of 1:20" is the heart of the problem — Tool #8 (Analyze the Units) keeps the bookkeeping honest: real feet × $\frac{1 model foot}{20 real feet}$ leaves the answer cleanly in model feet. Once we divide 289 ÷ 20 and round, Tool #3 (Eliminate Possibilities) lets us check the result against the five choices and spot (A) immediately, which is the standard AMC multiple-choice safety net.

1STEP 1

A 1:20 scale means 1 model foot per 20 real feet, so model height = real height ÷ 20.

model height =

\frac{real height}{20}

\frac{289 ft}{20}

2STEP 2

Dividing gives 289 ÷ 20 = 14.45 feet exactly (quotient 14, remainder 9, and $\frac{9}{20}$ = 0.45).

289 ÷ 20 = 14 R 9 → 14 +

\frac{9}{20}

= 14.45 ft

3STEP 3

The tenths digit is 4 (＜ 5), so 14.45 rounds down to 14 feet.

14.45 ≈ 14 (tenths digit 4 ＜ 5, round down)

4STEP 4

Only choice (A) equals 14; (B)–(E) don't match the rounded value, so they are eliminated.

14 = (A)

Answer

A 1:20 scale model of a 289-foot building should be roughly

\frac{1}{20}

of 289 ≈

\frac{300}{20}

= 15 feet — a height a person could stand next to. Our answer of 14 feet sits right next to that estimate and is small enough to fit in an amusement-park exhibit, so the magnitude is reasonable.

💡Key takeaway

This AMC 8 problem only needs Grade 6 ratio language — "1:20 means divide by 20" — you already know!

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