AMC 10 · 2013 · #1

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

ratedecimal-arithmeticdimensional-analysis identify-subproblems ↑ Prerequisites: rate

📏 Short solution 💡 1 insight

📘 View easy version →

Problem

A taxi ride costs $1.50$ plus$ 0.25 $per mile traveled. How much does a$ 5$-mile taxi ride cost?

Pick an answer.

(A)

2.25

(B)

2.50

(C)

2.75

(D)

3.00

(E)

3.75

AMC 10 2013 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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: A taxi charges a fixed starting fee of $1.50, then adds $0.25 for every mile driven. Find the total cost of a 5-mile ride.

Givens: Base fare is $1.50, charged once no matter the distance.; The per-mile rate is $0.25 for each mile traveled.; The ride is 5 miles long.

Unknowns: The total cost of the 5-mile ride.

Understand

Restated: A taxi charges a fixed starting fee of $1.50, then adds $0.25 for every mile driven. Find the total cost of a 5-mile ride.

Givens: Base fare is $1.50, charged once no matter the distance.; The per-mile rate is $0.25 for each mile traveled.; The ride is 5 miles long.

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

This is a rate problem with dollars and miles, so tracking what each number measures keeps the fixed fee separate from the per-mile charge. Splitting the total into a base part and a mileage part turns it into two easy calculations.

Execute — Answer: C

#8 Analyze the Units 5.OA.A.2 Step 1

Split the cost into two parts

The total is a fixed base fare plus a charge that grows with distance.
Write it as one starting fee of $1.50 added to $0.25 for each of the 5 miles.

\text{total} = \$1.50 + \$0.25 \times 5

💡 The dollars-per-mile only attaches to the miles, so the flat fee has to stand on its own.

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

Charge for the miles

Multiply the per-mile rate by the number of miles: $0.25 for each of 5 miles gives the mileage part of the fare.

\$0.25 \times 5 = \$1.25

💡 Five quarters is $1.25, the same as four quarters plus one more.

#8 Analyze the Units 5.NBT.B.7 Step 3

Add the base fare

Add the fixed $1.50 fee to the $1.25 mileage charge to get the total.
That total, $2.75, matches choice (C).

\$1.50 + \$1.25 = \$2.75

💡 Line up the cents columns and add: the two amounts combine cleanly to $2.75.

[1] #8 5.OA.A.2 The total is a fixed base fare plus a charge that grows with distance. Write it

[2] #7 5.NBT.B.7 Multiply the per-mile rate by the number of miles: $0.25 for each of 5 miles giv

[3] #8 5.NBT.B.7 Add the fixed $1.50 fee to the $1.25 mileage charge to get the total. That total

Review

Reasonableness: A short 5-mile ride should cost a little more than the base fare of $1.50, and $2.75 is only $1.25 above it, which fits five quarters of driving. The answer sits sensibly between the cheapest and most expensive choices.

Alternative: Convert to cents: 150 + 25 x 5 = 150 + 125 = 275 cents = $2.75. Working in whole cents avoids decimals and gives the same result (C).

CCSS standards used (min grade 5)

5.OA.A.2 Write simple expressions that record calculations with numbers (Setting up total cost as base fare plus rate times miles.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying $0.25 by 5 and adding the $1.50 base fare.)

⭐ Charge the flat fee once, then add the per-mile rate times the miles.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Split the cost into two parts

Charge for the miles

Add the base fare

Review

CCSS standards used (min grade 5)

More like this