2000 AMC 10 Problem 4

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 $\boxed{\textbf{(D)}\ \$7.42}$.

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.

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.

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 $\boxed{\textbf{(D)}\ \$7.42}$.