AMC 8 · 2011 · #10

Easy mode Grade 6

Problem

In Gotham City, a taxi works like this. The first half mile costs $\$2.40$ no matter what. After that, every extra $0.1$ mile adds another $\$0.20$ .

You have $\$10$ to spend in total. You plan to give the driver a $\$2$ tip out of that.

How many miles can you ride with the money you have left for the fare?

Pick an answer.

(A)

3.0

(B)

3.25

(C)

3.3

(D)

3.5

(E)

3.75

AMC 8 2011 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.

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Toolkit + CCSS Solution

Understand

Restated: A Gotham City taxi charges $\$2.40$ for the first $\tfrac{1}{2}$ mile, then $\$0.20$ for each additional $0.1$ mile. You have $\$10$ total and want to leave a $\$2$ tip. How many miles can you ride?

Givens: Flat fare = $\$2.40$ covers the first $0.5$ mile; After $0.5$ mile, the rate is $\$0.20$ per $0.1$ mile; Total money available = $\$10$; Planned tip = $\$2$; Answer choices: (A) $3.0$, (B) $3.25$, (C) $3.3$, (D) $3.5$, (E) $3.75$ (miles)

Unknowns: The total number of miles the ride can cover under the $\$10$ budget

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

The $\$10$ has to do three different jobs — tip, flat fare for the first $0.5$ mile, and the per-$0.1$-mile charge after that — so Tool #7 (Identify Subproblems) cleanly peels them off one at a time: take out the tip, take out the flat fare, then ask "how far does the leftover money buy?" Tool #8 (Analyze the Units) turns the awkward $\$0.20$ per $0.1$ mile rate into the friendlier $\$2$ per mile, which makes the final division a one-line decimal computation rather than an algebra problem.

Execute — Answer: C

#7 Identify Subproblems 4.MD.A.2 Step 1

Take out the tip

Subproblem 1 — set aside the tip.
You will hand the driver $\$2$ at the end, so the meter can only use what is left.

$\$10.00 - \$2.00 = \$8.00 \text{ for the meter}$

💡 Splitting the $\$10$ into "tip" and "fare" is the Tool #7 move — handle one job at a time.

#7 Identify Subproblems 5.NBT.B.7 Step 2

Pay the flat fare

Subproblem 2 — pay the flat fare for the first $\tfrac{1}{2}$ mile.
Subtract it from the meter budget; whatever remains has to cover the rest of the trip.

$\$8.00 - \$2.40 = \$5.60 \text{ left for extra distance}$

💡 Subtracting decimals to the hundredths place is a Grade 5 arithmetic move.

#8 Analyze the Units 6.RP.A.3 Step 3

Rewrite the rate per mile

Convert the meter's rate into a friendlier unit.
$\$0.20$ per $0.1$ mile means every full mile costs ten times $\$0.20$, i.e.
$\$2.00$ per mile.

$\dfrac{\$0.20}{0.1 \text{ mile}} = \dfrac{\$0.20 \times 10}{0.1 \times 10 \text{ mile}} = \dfrac{\$2.00}{1 \text{ mile}}$

💡 Scaling the top and bottom of the rate by $10$ turns "per $0.1$ mile" into "per mile" — a Grade 6 unit-rate move.

#7 Identify Subproblems 5.NBT.B.7 Step 4

Divide to get extra miles

Subproblem 3 — figure out how many extra miles the $\$5.60$ buys at $\$2.00$ per mile.

$\dfrac{\$5.60}{\$2.00 / \text{mile}} = 2.8 \text{ miles of extra distance}$

💡 Dividing $5.60$ by $2$ is a direct Grade 5 decimal calculation.

#7 Identify Subproblems 5.NBT.B.7 Step 5

Add back the first half mile

Add the flat-fare distance back to the extra distance to get the full ride length.

$$0.5 + 2.8 = 3.3 \text{ miles} \;\Rightarrow\; \textbf{(C)}$$

💡 Re-combining the two subproblem pieces (first $0.5$ mile $+$ extra $2.8$ mile) finishes the Tool #7 split.

[1] #7 4.MD.A.2 Subproblem 1 — set aside the tip. You will hand the driver $\$2$ at the end, so

[2] #7 5.NBT.B.7 Subproblem 2 — pay the flat fare for the first $\tfrac{1}{2}$ mile. Subtract it

[3] #8 6.RP.A.3 Convert the meter's rate into a friendlier unit. $\$0.20$ per $0.1$ mile means e

[4] #7 5.NBT.B.7 Subproblem 3 — figure out how many extra miles the $\$5.60$ buys at $\$2.00$ per

[5] #7 5.NBT.B.7 Add the flat-fare distance back to the extra distance to get the full ride lengt

Review

Reasonableness: Check the dollars: first $0.5$ mile costs $\$2.40$, the extra $2.8$ miles cost $2.8 \times \$2.00 = \$5.60$, and the tip is $\$2.00$. Total $= 2.40 + 5.60 + 2.00 = \$10.00$ exactly — every dollar is accounted for, so $3.3$ miles is right. A quick sanity check on size: at $\$2$ per mile after the first half, an extra $\$5.60$ should buy under $3$ extra miles, so the total ride should be a bit under $3.5$ miles. $3.3$ fits perfectly.

Alternative: Tool #6 (Guess and Check) on the choices. For each candidate distance $x$, the total cost is $\$2.40 + (x - 0.5) \times \$2.00 + \$2.00 \text{ tip}$, and we need this to equal $\$10.00$. (A) $3.0$: $2.40 + 2.5 \times 2 + 2 = \$9.40$ (too cheap). (B) $3.25$: $2.40 + 2.75 \times 2 + 2 = \$9.90$ (still cheap). (C) $3.3$: $2.40 + 2.8 \times 2 + 2 = \$10.00$ — exact match. (D) $3.5$ and (E) $3.75$ overshoot $\$10$, so (C) is the answer.

CCSS standards used (min grade 6)

4.MD.A.2 Solve word problems involving distances, time, liquid volumes, masses, and money (Setting aside the $\$2$ tip from the $\$10$ total ($\$10 - \$2 = \$8$) as a money word-problem subtraction.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (The decimal arithmetic in the body of the solution: $\$8.00 - \$2.40 = \$5.60$, $\$5.60 \div \$2.00 = 2.8$, and $0.5 + 2.8 = 3.3$.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Rewriting the meter rate $\$0.20$ per $0.1$ mile as the equivalent unit rate $\$2.00$ per mile, then using that rate to convert the remaining $\$5.60$ into a distance.)

⭐ Big AMC 8 word problems often shrink down to Grade 6 rate reasoning — once you split off the tip and the flat fare, only a tidy decimal division is left.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Take out the tip

Pay the flat fare

Rewrite the rate per mile

Divide to get extra miles

Add back the first half mile

Review

CCSS standards used (min grade 6)

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