AMC 8 · 2018 · #1

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

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

📏 Short solution 💡 2 insights

Problem

An amusement park has a collection of scale models, with a ratio of $1: 20$ , of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number?

$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }18\qquad\textbf{(E) }20$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: An amusement park makes scale models at a $1:20$ ratio, meaning every model is $\tfrac{1}{20}$ 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?

Givens: Scale ratio model$:$real $= 1:20$; Real height of the U.S. Capitol $= 289$ feet; Answer must be rounded to the nearest whole foot; Answer choices: (A) $14$, (B) $15$, (C) $16$, (D) $18$, (E) $20$

Unknowns: The height of the model (in feet, rounded to the nearest whole number)

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #3 Eliminate Possibilities

The phrase "ratio of $1:20$" is the heart of the problem — Tool #8 (Analyze the Units) keeps the bookkeeping honest: real feet $\times \tfrac{1 \text{ model foot}}{20 \text{ real feet}}$ leaves the answer cleanly in model feet. Once we divide $289 \div 20$ 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.

Execute — Answer: A

#8 Analyze the Units 6.RP.A.1 Step 1

Translate the ratio into an operation.
A $1:20$ scale means $1$ foot of model for every $20$ feet of real building, so the model height is the real height divided by $20$.

$$\text{model height} = \dfrac{\text{real height}}{20} = \dfrac{289 \text{ ft}}{20}$$

💡 Reading $1:20$ as "$1$ part model per $20$ parts real" is exactly the Grade 6 ratio-language idea.

#8 Analyze the Units 5.NBT.B.6 Step 2

Carry out the division of $289$ by $20$.
Since $20 \times 14 = 280$, the quotient is $14$ with a remainder of $9$, and $\tfrac{9}{20} = 0.45$, so $289 \div 20 = 14.45$ feet exactly.

$$289 \div 20 = 14 \text{ R } 9 \;\Rightarrow\; 14 + \tfrac{9}{20} = 14.45 \text{ ft}$$

💡 Dividing a three-digit number by a two-digit number with a remainder is a Grade 5 standard long-division skill.

#8 Analyze the Units 5.NBT.A.4 Step 3

Round $14.45$ to the nearest whole number.
The digit in the tenths place is $4$, which is less than $5$, so we round down to $14$.

$14.45 \approx 14$ (tenths digit $4 < 5$, round down)

💡 Rounding a decimal to the ones place by looking at the tenths digit is the Grade 5 "round decimals to any place" rule.

#3 Eliminate Possibilities 5.NBT.A.4 Step 4

Match $14$ feet to the answer choices.
Only choice (A) equals $14$; (B)–(E) are all larger than the computed $14.45$ would round to, so they are eliminated.

$$14 = \textbf{(A)}$$

💡 Once the rounded value is fixed at $14$, the only choice that matches is (A) — the other four are ruled out by inspection.

[1] #8 6.RP.A.1 Translate the ratio into an operation. A $1:20$ scale means $1$ foot of model fo

[2] #8 5.NBT.B.6 Carry out the division of $289$ by $20$. Since $20 \times 14 = 280$, the quotien

[3] #8 5.NBT.A.4 Round $14.45$ to the nearest whole number. The digit in the tenths place is $4$,

[4] #3 5.NBT.A.4 Match $14$ feet to the answer choices. Only choice (A) equals $14$; (B)–(E) are

Review

Reasonableness: A $1:20$ scale model of a $289$-foot building should be roughly $\tfrac{1}{20}$ of $289 \approx \tfrac{300}{20} = 15$ 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.

Alternative: Tool #6 (Guess and Check) on the choices: multiply each candidate by $20$ to recover the implied real height. (A) $14 \times 20 = 280$, (B) $15 \times 20 = 300$, (C) $16 \times 20 = 320$, (D) $18 \times 20 = 360$, (E) $20 \times 20 = 400$. The real height $289$ sits between $280$ and $300$, closer to $280$ (only $9$ above) than to $300$ ($11$ below), so the rounded model height is $14$ — choice (A).

CCSS standards used (min grade 6)

5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors (Computing $289 \div 20 = 14$ remainder $9$, then expressing the remainder as $\tfrac{9}{20} = 0.45$ to get the exact model height $14.45$ feet.)
5.NBT.A.4 Round decimals to any place (Rounding $14.45$ to the nearest whole number by checking the tenths digit and rounding down to $14$.)
6.RP.A.1 Understand the concept of a ratio and use ratio language (Translating the phrase "ratio of $1:20$" into the operation "model height $= $ real height $\div 20$.")

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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