2013 AMC 10A Problem 1

P1-10~1 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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

Show solution without answering

Key insight

The 1.50 is paid once, but the 0.25 is paid again for every mile, so add 1.50 to 5 times 0.25.

Solution

The price has two parts, so work them out one at a time.

1. The part you always pay. Getting in the taxi costs $\$1.50$ , no matter how far you go.
2. The part that grows with distance. Every mile adds $\$0.25$ . For $5$ miles that is
$5 \times 0.25 = 1.25 \text{ dollars}.$
3. Add the two parts.
$1.50 + 1.25 = 2.75.$

A quick check: $\$0.25$ is a quarter, and $5$ quarters is $\$1.25$ , which matches Step 2.

The ride costs $\boxed{\textbf{(C)}\ 2.75}$ .

Why this works

Many word problems have the shape "a fixed amount plus so much for each unit": taxi fares, phone plans, bike rentals. Find the number that is paid once and the number that gets multiplied by the quantity. Then multiply and add. A cost of $1.50 + 0.25 \times (\text{miles})$ is an example of a linear formula, and this problem only asks you to put in $5$ miles.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The ride has two parts: a flat charge of $\$1.50$ that is paid regardless of distance, and a distance charge of $\$0.25$ for every mile.

Five miles cost $5 \times 0.25 = 1.25$ dollars. Adding the flat charge:
$1.50 + 1.25 = 2.75.$

The ride costs $\boxed{\textbf{(C)}\ 2.75}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)