A certain taxi company charges $3.10 for the first 1/5 of a mile plus $0.40 for each additional 1/5 of...

1. Translate the problem requirements: The taxi charges $3.10 for the first \(\frac{1}{5}\) mile, then $0.40 for each additional \(\frac{1}{5}\) mile segment. We need to find the total cost for an 8-mile ride.
2. Convert the total distance into rate segments: Figure out how many \(\frac{1}{5}\)-mile segments make up 8 miles to understand the charging structure.
3. Separate the base charge from additional charges: Identify what falls under the initial $3.10 charge versus what gets charged at the $0.40 rate.
4. Calculate the additional segment charges: Determine how many segments are charged at $0.40 each and compute that cost.
5. Sum up the total fare: Add the base charge and additional charges to get the final taxi fare.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the taxi company charges in simple terms:

• There's a base charge of $3.10 that covers the first \(\frac{1}{5}\) of a mile
• After that first \(\frac{1}{5}\) mile, every additional \(\frac{1}{5}\) mile costs $0.40
• We need to find the total cost for an 8-mile ride

Think of it like a parking meter: you pay a certain amount upfront, then additional amounts for each extra time period.

Process Skill: TRANSLATE - Converting the rate structure into clear mathematical understanding

2. Convert the total distance into rate segments

Since the company charges by \(\frac{1}{5}\)-mile segments, let's figure out how many of these segments are in 8 miles.

In everyday terms: How many \(\frac{1}{5}\)s fit into 8?

To find this, we divide: \(8 \div \frac{1}{5} = 8 \times 5 = 40\)

So an 8-mile ride consists of 40 segments of \(\frac{1}{5}\) mile each.

3. Separate the base charge from additional charges

Now we need to understand what gets charged at which rate:

• The FIRST \(\frac{1}{5}\)-mile segment is covered by the $3.10 base charge
• ALL the REMAINING segments get charged at $0.40 each

Since we have 40 total segments, and the first one is covered by the base charge:

Additional segments that cost $0.40 each = \(40 - 1 = 39\) segments

4. Calculate the additional segment charges

Now we can calculate the cost for those additional segments:

Additional segments: 39

Cost per additional segment: $0.40

Total additional charges = \(39 \times \$0.40 = \$15.60\)

5. Sum up the total fare

Finally, we add up all the charges:

• Base charge (first \(\frac{1}{5}\) mile): $3.10
• Additional charges (39 segments): $15.60
• Total fare = \(\$3.10 + \$15.60 = \$18.70\)

Final Answer

The taxi company would charge $18.70 for an 8-mile ride.

This matches answer choice D: $18.70

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the rate structure: Students may think that the $3.10 charge is separate from the first \(\frac{1}{5}\) mile, leading them to charge $3.10 + $0.40 for the first segment instead of recognizing that $3.10 already covers the first \(\frac{1}{5}\) mile.

2. Confusion about fractional mile units: Students might struggle with the concept of charging by \(\frac{1}{5}\)-mile segments and attempt to work directly with the 8-mile distance without converting it into the appropriate billing units.

3. Misinterpreting "additional" segments: Students may incorrectly assume that all 40 segments should be charged at $0.40 each, failing to recognize that "additional" means segments beyond the first one that's already covered by the base charge.

Errors while executing the approach

1. Incorrect division when converting miles to segments: Students might calculate \(8 \div \frac{1}{5}\) incorrectly, perhaps getting \(\frac{8}{5} = 1.6\) instead of \(8 \times 5 = 40\), due to confusion with fraction division rules.

2. Wrong count of additional segments: Even after correctly finding 40 total segments, students might use 40 instead of 39 when calculating additional charges, forgetting to subtract the first segment that's covered by the base charge.

3. Arithmetic errors in multiplication: Students may make calculation mistakes when computing \(39 \times \$0.40\), potentially getting $15.80 or $14.60 instead of the correct $15.60.

Errors while selecting the answer

1. Adding components incorrectly: Students might add their calculated values in the wrong combination, such as forgetting to include the base charge and only selecting the additional charges ($15.60) as their final answer.

2. Rounding or decimal place errors: Students may arrive at the correct approach but make small errors in their final addition, leading them to select a nearby answer choice like $18.60 or $18.80 instead of $18.70.