AMC 8 · 2004 · #1
Grade 6 arithmeticProblem
On a map, a 12-centimeter length represents 72 kilometers. How many kilometers does a 17-centimeter length represent?
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
6.EE.A.2 Step 1 - Name the unknown.
- Let $x$ be the number of kilometers that $17$ cm on the map represents.
💡 Giving the unknown a letter is the Grade 6 "write expressions with variables" move.
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.
💡 Dividing the two quantities turns a $12$-to-$72$ ratio into the per-centimeter rate the map uses everywhere.
6.RP.A.3 Step 3 - Apply the rate to $17$ centimeters.
- The same $6$ km/cm rate works for every length, so multiply.
💡 Once you know the per-unit value, scaling up is one multiplication.
6.RP.A.3 Step 4 - Match the value to a choice.
- $102$ km is answer (B).
💡 The unit-rate method always gives a number that matches one of the choices exactly when the problem is well posed.
6.EE.A.2 Name the unknown. Let $x$ be the number of kilometers that $17$ cm on the map re 6.RP.A.2 Find the unit rate. Since $12$ cm corresponds to $72$ km, divide to see how many 6.RP.A.3 Apply the rate to $17$ centimeters. The same $6$ km/cm rate works for every leng 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.2Write, 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.2Understand 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.3Use 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.
⭐ Map scale problems become easy once you pin down the per-centimeter rate — find it once, then multiply by whatever length you need.
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