2000 AMC 10 Problem 4

P1-10~2 minPrint

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

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $\$12.48$ , but in January her bill was $\$17.54$ because she used twice as much connect time as in December. What is the fixed monthly fee?

Show solution without answering

Key insight

January used exactly one extra December's worth of connect time, so the difference between the two bills is the cost of December's time.

Solution

1. Each bill has two parts: the fixed fee, which is the same every month, and the charge for time online.
2. Call December's time charge "one block." Then:
- December's bill $=$ fee $+$ one block $= \$12.48$ .
- January used twice the time, so January's bill $=$ fee $+$ two blocks $= \$17.54$ .
3. The two bills have the same fee, so the only difference is the extra block. One block costs
$17.54 - 12.48 = 5.06 \text{ dollars}.$
4. Now go back to December: fee $= 12.48 - 5.06 = 7.42$ dollars.
5. Check January: $7.42 + 2 \cdot 5.06 = 7.42 + 10.12 = 17.54$ . It works.

The answer is $\boxed{\textbf{(D)}\ \$7.42}$ .

Why this works

A bill that is "fixed part plus a part for each hour" changes only in the hourly part. Comparing two bills cancels the fixed part and leaves just the extra hours. Once you know the hourly part, the fixed part is what is left over. Watch out for which of the two you were asked for.

Alternative approach

Test the choices. For each fee, December's time costs $12.48$ minus the fee, and January's should be double that. With (D), December's time is $5.06$ , January's is $10.12$ , and $7.42 + 10.12 = 17.54$ . Only (D) matches January's bill.

Concise solution

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

Let $F$ be the fixed fee and $C$ the cost of December's connect time. December: $F + C = 12.48$ . January used twice the time, so $F + 2C = 17.54$ .

Subtracting the first equation from the second gives $C = 17.54 - 12.48 = 5.06$ . Then $F = 12.48 - 5.06 = 7.42$ .

The fixed monthly fee is $\boxed{\textbf{(D)}\ \$7.42}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 5: Multi-Variable Linear Equations (§5.2, §5.3, §5.6)
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Intermediate Algebra, Chapter 1: Basic Techniques for Solving Equations (§1.4)