Question

A taxi driver charges for a ride at a rate of 2 dollars for the first half mile and 75 cents for each additional half-mile. Which of the following expressions represents the total charge, in cents for \[p\] miles, where \[p\] is a positive integer?
A. \[25\left( {6p + 8} \right)\]
B. \[25\left( {6p + 5} \right)\]
C. \[25\left( {3p + 5} \right)\]
D. \[1.25 + 1.5p\]
E. \[2 + 0.75\left( {p - 1} \right)\]

Answer 1

Hint: First of all, calculate the charge for the first half-mile in cents. Then find out the remaining number of miles by subtracting a half-mile from the total number of miles i.e., \[p\] miles. Further, calculate the charge for these remaining numbers of miles. Add up both the charges to get the total fare for \[p\] miles. So, use this concept to reach the solution to the given problem.

Complete step-by-step solution:
Given that the charge for a ride in a certain city is 2 dollars for the first half of a mile and 75 cents for each additional half of a mile.
We have to calculate the charge for \[p\] miles.
The charge for first half mile = 2 dollars = 200 cents
The charge for the next half miles = 75 cents
Or
The charge for next one mile = \[2 \times 75 = 150\] cents
The remaining number of miles in \[p\] miles of distance after the first half mile = \[\left( {p - \dfrac{1}{2}} \right)\] miles.
The charge for the remaining \[\left( {p - \dfrac{1}{2}} \right)\] miles = \[\left( {p - \dfrac{1}{2}} \right) \times 150\] cents
Therefore, the total charge for the \[p\] miles = charge for first half mile + charge for \[\left( {p - \dfrac{1}{2}} \right)\] miles
                                                                               \[
   = 200 + \left( {p - \dfrac{1}{2}} \right) \times 150 \\
   = 200 + 150p - \dfrac{{150}}{2} \\
   = 200 + 150p - 75 \\
   = 125 + 150p \\
   = 25\left( {6p + 5} \right){\text{ cents}} \\
\]

Thus, the correct option is B. \[25\left( {6p + 5} \right)\]

Note: One dollar is equal to hundred cents. In this problem we have converted the charge for the next half miles in to charge for the next one mile. We can also solve this by converting the remaining \[\left( {p - \dfrac{1}{2}} \right)\] miles in to number of half miles i.e., \[\dfrac{{\left( {p - \dfrac{1}{2}} \right)}}{2}\] miles. Both the methods give the same answer.