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A certain taxi company charges \(\$3.10\) for the first \(\frac{1}{5}\) of a mile plus \(\$0.40\) for each additional \(\frac{1}{5}\) of a mile. What would this company charge for a taxi ride that was \(8\) miles long?
Let's break down what the taxi company charges in simple terms:
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
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.
Now we need to understand what gets charged at which rate:
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
Now we can calculate the cost for those additional segments:
Additional segments: 39
Cost per additional segment: $0.40
Total additional charges = \(39 \times \$0.40 = \$15.60\)
Finally, we add up all the charges:
The taxi company would charge $18.70 for an 8-mile ride.
This matches answer choice D: $18.70
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.
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.
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.