AMC 8 · 2004 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

ratio-proportionrateunit-conversion identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

Problem

On a map, a $12$ -centimeter length represents $72$ kilometers. How many kilometers does a $17$ -centimeter length represent?

Pick an answer.

(A)

(B)

102

(C)

204

(D)

864

(E)

1224

View mode:

Toolkit + CCSS Solution

Understand

Restated: On a map, $12$ centimeters stands for $72$ kilometers. How many kilometers does a $17$-centimeter length on the same map represent?

Givens: $12$ cm on the map represents $72$ km in reality; We want the real distance represented by $17$ cm on the same map; Answer choices: (A) $6$, (B) $102$, (C) $204$, (D) $864$, (E) $1224$

Unknowns: The number of kilometers represented by $17$ centimeters on the map

Understand

Restated: On a map, $12$ centimeters stands for $72$ kilometers. How many kilometers does a $17$-centimeter length on the same map represent?

Givens: $12$ cm on the map represents $72$ km in reality; We want the real distance represented by $17$ cm on the same map; Answer choices: (A) $6$, (B) $102$, (C) $204$, (D) $864$, (E) $1224$

Plan

Primary tool: #3 Set Up an Equation

Secondary: #4 Introduce a Variable

A map scale is a fixed ratio: kilometers per centimeter is the same for every length on the map. That is exactly the setup for Tool #3 (Set Up an Equation) — write a proportion that says "$12$ cm matches $72$ km the same way $17$ cm matches $x$ km." Tool #4 (Introduce a Variable) names the unknown distance $x$ so we can solve. The cleanest path is to find the unit rate (km per cm) first, then multiply by $17$.

Execute — Answer: B

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the unknown.
Let $x$ be the number of kilometers that $17$ cm on the map represents.

$$x = \text{km represented by } 17 \text{ cm}$$

💡 Giving the unknown a letter is the Grade 6 "write expressions with variables" move.

#3 Set Up an Equation 6.RP.A.2 Step 2

Find the unit rate.
Since $12$ cm corresponds to $72$ km, divide to see how many kilometers go with a single centimeter.

$$\text{rate} = \dfrac{72 \text{ km}}{12 \text{ cm}} = 6 \text{ km/cm}$$

💡 Dividing the two quantities turns a $12$-to-$72$ ratio into the per-centimeter rate the map uses everywhere.

#3 Set Up an Equation 6.RP.A.3 Step 3

Apply the rate to $17$ centimeters.
The same $6$ km/cm rate works for every length, so multiply.

$$x = 17 \text{ cm} \times 6 \text{ km/cm} = 102 \text{ km}$$

💡 Once you know the per-unit value, scaling up is one multiplication.

#3 Set Up an Equation 6.RP.A.3 Step 4

Match the value to a choice.
$102$ km is answer (B).

$$x = 102 \;\Rightarrow\; \textbf{(B)}$$

💡 The unit-rate method always gives a number that matches one of the choices exactly when the problem is well posed.

[1] #4 6.EE.A.2 Name the unknown. Let $x$ be the number of kilometers that $17$ cm on the map re

[2] #3 6.RP.A.2 Find the unit rate. Since $12$ cm corresponds to $72$ km, divide to see how many

[3] #3 6.RP.A.3 Apply the rate to $17$ centimeters. The same $6$ km/cm rate works for every leng

[4] #3 6.RP.A.3 Match the value to a choice. $102$ km is answer (B).

Review

Reasonableness: Quick sanity check: $17$ cm is a little more than $12$ cm, so the real distance should be a little more than $72$ km. Our answer $102$ km fits that — it is bigger than $72$ but nowhere near the much larger choices $204$, $864$, or $1224$. Also, $102 \div 17 = 6$ recovers the same unit rate we found, so the scale is consistent.

Alternative: Tool #3 with a direct proportion: write $\dfrac{12}{72} = \dfrac{17}{x}$ and cross-multiply to get $12x = 72 \times 17 = 1224$, so $x = 1224 \div 12 = 102$ km. Same answer (B), no unit rate computed separately. The wrong choice (E) $1224$ is exactly the cross-product before dividing — a useful reminder to finish the proportion.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the unknown real distance with the variable $x$ so the proportion can be written algebraically.)
6.RP.A.2 Understand the concept of a unit rate associated with a ratio (Turning the given $12$ cm to $72$ km ratio into the unit rate $6$ km per cm that the map scale represents.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Applying the $6$ km/cm rate to a new length of $17$ cm to find the corresponding real distance.)

⭐ Map scale problems become easy once you pin down the per-centimeter rate — find it once, then multiply by whatever length you need.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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