For the rental of a certain type of car, rental agency R charges a fee of $30 per day plus...

1. Translate the problem requirements: We need to find the total miles where Agency R's cost (daily fee + excess mileage charges) equals Agency S's cost (flat daily fee) over 3 days
2. Set up cost expressions for each agency: Express the total 3-day cost for each agency in terms of total miles driven
3. Establish the break-even equation: Set the two cost expressions equal to find when costs are identical
4. Solve for total mileage: Isolate the total miles variable through algebraic manipulation

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what each rental agency charges in plain English:

Agency R: Think of this as a "base + overage" plan. You pay \(\$30\) per day, and if you drive more than 100 miles in a day, you pay an extra \(\$0.20\) for every mile over that limit.

Agency S: This is the "all-inclusive" plan. You pay \(\$65\) per day and can drive as much as you want with no extra charges.

Since we're renting for 3 days, we need to find the total number of miles where both agencies cost exactly the same amount.

Process Skill: TRANSLATE - Converting the problem's rental terms into mathematical understanding

2. Set up cost expressions for each agency

Let's say the total miles driven over 3 days is M miles. Since the car is driven the same number of miles each day, that means \(\mathrm{M/3}\) miles per day.

Agency R's total cost:

• Base cost: \(\$30 \times 3 \text{ days} = \$90\)
• Extra mileage cost: Only if daily miles exceed 100
• If \(\mathrm{M/3 > 100}\) (meaning \(\mathrm{M > 300}\)), then excess miles per day = \(\mathrm{(M/3) - 100}\)
• Total excess miles over 3 days = \(\mathrm{3 \times [(M/3) - 100] = M - 300}\)
• Extra cost = \(\mathrm{(M - 300) \times \$0.20}\)
• Total cost for Agency R = \(\mathrm{\$90 + \$0.20(M - 300)}\)

Agency S's total cost:

• Simple: \(\mathrm{\$65 \times 3 \text{ days} = \$195}\)

Since we're looking at answer choices all above 300 miles, we know \(\mathrm{M > 300}\), so our Agency R formula applies.

3. Establish the break-even equation

At the break-even point, both agencies cost the same amount. So we set them equal:

Agency R cost = Agency S cost
\(\mathrm{\$90 + \$0.20(M - 300) = \$195}\)

This equation tells us: "When does the base-plus-overage plan cost the same as the all-inclusive plan?"

4. Solve for total mileage

Now let's solve step by step:

\(\mathrm{\$90 + \$0.20(M - 300) = \$195}\)

First, let's expand the left side:
\(\mathrm{\$90 + \$0.20M - \$60 = \$195}\)
\(\mathrm{\$30 + \$0.20M = \$195}\)

Subtract \(\$30\) from both sides:
\(\mathrm{\$0.20M = \$165}\)

Divide both sides by \(\$0.20\):
\(\mathrm{M = \$165 ÷ \$0.20 = 825}\)

Let's verify this makes sense:

• Daily miles: \(\mathrm{825 ÷ 3 = 275}\) miles per day
• Agency R: \(\mathrm{\$90 + \$0.20(825 - 300) = \$90 + \$0.20(525) = \$90 + \$105 = \$195}\)
• Agency S: \(\mathrm{\$65 \times 3 = \$195}\) ✓

Final Answer

The total number of miles where both agencies cost the same is 825 miles, which corresponds to answer choice E.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the "excess miles" constraint

Students often miss that Agency R only charges extra for miles in excess of 100 miles per day, not for all miles driven. They might set up the cost equation as \(\mathrm{\$30 \times 3 + \$0.20 \times M}\) instead of correctly accounting for the 100-mile daily threshold. This leads to an incorrect equation that doesn't reflect the actual pricing structure.

2. Confusion about daily vs. total mileage calculations

The problem states the car will be driven "the same number of miles each day" for 3 days. Students may struggle to correctly express that if M is total miles, then \(\mathrm{M/3}\) is daily miles, and excess daily miles are \(\mathrm{(M/3 - 100)}\). Some students incorrectly use \(\mathrm{M - 300}\) as daily excess instead of understanding it represents total excess over 3 days.

3. Forgetting to check if the excess mileage condition applies

Students might automatically assume excess charges apply without verifying that daily mileage exceeds 100 miles. They should recognize that if \(\mathrm{M ≤ 300}\) (meaning ≤100 miles per day), Agency R would simply charge \(\$90\) total with no excess fees, making the comparison straightforward.

Errors while executing the approach

1. Arithmetic errors in equation setup and expansion

When expanding \(\mathrm{\$90 + \$0.20(M - 300)}\), students commonly make errors like forgetting to distribute the \(\$0.20\) to both terms, incorrectly calculating \(\mathrm{\$0.20 \times 300 = \$60}\), or making sign errors when simplifying to get \(\mathrm{\$30 + \$0.20M = \$195}\).

2. Division errors when solving for M

The final step requires dividing \(\$165\) by \(\$0.20\). Students often struggle with decimal division, potentially calculating \(\mathrm{165 ÷ 0.2}\) incorrectly. Converting to \(\mathrm{1650 ÷ 2 = 825}\) or recognizing that dividing by 0.2 is the same as multiplying by 5 can help avoid this error.

Errors while selecting the answer

1. Selecting daily mileage instead of total mileage

After calculating \(\mathrm{M = 825}\) total miles, students might verify by finding daily miles \(\mathrm{(825 ÷ 3 = 275 \text{ miles per day})}\) and then mistakenly look for 275 among the answer choices. The question asks for "total number of miles," not daily miles, so 825 is correct.