A certain state levies a 4% tax on the nightly rates of hotel rooms. A certain hotel in this state...

1. Translate the problem requirements: Break down what constitutes the total charge - we have a nightly room rate (unknown), a 4% tax applied only to the room rate, and a fixed $2.00 fee not subject to tax. The sum of all these components equals $74.80, and we need to find the original nightly rate.
2. Set up the cost equation: Express the total charge as the sum of three components - the nightly rate (let's call it R), the tax on that rate (0.04R), and the non-taxable fee ($2.00).
3. Solve for the nightly rate: Use the equation \(\mathrm{R} + 0.04\mathrm{R} + 2.00 = 74.80\) to find R by combining like terms and isolating the variable.
4. Verify using answer choices: Check that our calculated answer matches one of the given options and confirms our solution logic.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what happens when someone stays at this hotel for one night. Think of it like a restaurant bill where you have the food cost, tax on the food, and a service charge.

Here we have:

• The nightly room rate (this is what we need to find)
• A 4% tax that gets applied ONLY to the room rate (not the fee)
• A fixed $2.00 fee that is NOT taxed
• The total of all these parts equals $74.80

So if someone pays $74.80 total, that $74.80 includes the room rate + tax on room rate + the $2 fee.

Process Skill: TRANSLATE - Converting the problem description into clear mathematical components

2. Set up the cost equation

Now let's express this in plain English first, then write it mathematically.

In everyday terms: "Room rate + Tax on room rate + Non-taxable fee = Total charge"

Let's call the room rate 'R'. Then:

• The room rate = R
• The tax on the room rate = 4% of R = \(0.04 \times \mathrm{R}\)
• The non-taxable fee = $2.00
• Total charge = $74.80

So our equation becomes: \(\mathrm{R} + 0.04\mathrm{R} + 2.00 = 74.80\)

3. Solve for the nightly rate

Now we solve this step by step:

\(\mathrm{R} + 0.04\mathrm{R} + 2.00 = 74.80\)

First, let's combine the R terms. Think of R as 1.00R, so:

\(1.00\mathrm{R} + 0.04\mathrm{R} = 1.04\mathrm{R}\)

Our equation becomes:

\(1.04\mathrm{R} + 2.00 = 74.80\)

Now subtract $2.00 from both sides:

\(1.04\mathrm{R} = 74.80 - 2.00\)

\(1.04\mathrm{R} = 72.80\)

Finally, divide both sides by 1.04:

\(\mathrm{R} = 72.80 \div 1.04\)

\(\mathrm{R} = 70.00\)

So the nightly room rate is $70.00.

4. Verify using answer choices

Let's check our answer by working forwards with R = $70.00:

• Room rate: $70.00
• Tax on room rate: \(4\% \times \$70.00 = 0.04 \times \$70.00 = \$2.80\)
• Non-taxable fee: $2.00
• Total: \(\$70.00 + \$2.80 + \$2.00 = \$74.80\) ✓

This matches the given total charge, confirming our answer.

Looking at the answer choices, $70.00 corresponds to choice (C).

Final Answer

The nightly rate of the room was $70.00, which is answer choice (C).

Common Faltering Points

Errors while devising the approach

Students often incorrectly assume that the 4% tax applies to the entire $74.80 total charge. They miss the crucial detail that the tax only applies to the nightly room rate, not the $2.00 fee. This leads them to set up an equation like: Total = Room rate + 4% of $74.80 + $2.00, which is fundamentally wrong.

Some students incorrectly think they should add the $2.00 fee to the room rate first, then apply the 4% tax to that sum. They set up: \(\text{Total} = (\text{Room rate} + \$2.00) \times 1.04\), rather than understanding that tax is calculated on room rate alone, then the fee is added separately.

Errors while executing the approach

When solving \(\mathrm{R} + 0.04\mathrm{R} + 2.00 = 74.80\), students frequently make errors combining R and 0.04R. They might incorrectly get 0.04R or 1.4R instead of the correct 1.04R, leading to wrong final answers.

When dividing 72.80 by 1.04, students often struggle with decimal division and may get incorrect results like $70.77 or round incorrectly. Some may also forget to perform this final division step altogether.

Errors while selecting the answer

Students might calculate correctly that 1.04R = 72.80, but then mistakenly select $72.00 (answer choice E) thinking this represents the room rate, rather than completing the division to get $70.00. They confuse the intermediate step with the final answer.