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Chapter 1: Problem 11

A fair fare. Three strangers, Bob, Mary, and Ivan, meet at a taxi stand and decide to share a cab to cut down the cost. Each has a different destination, but all the destinations are on the highway leading from the airport, so no circuitous driving is required. Bob's destination is 10 miles away, Mary's is 20 miles, and Ivan's is 30 miles. The taxi costs \(\$ 1.50\) per mile including the tip, regardless of the number of passengers. How much should each person pay? (Caution: There is more than one way of looking at this situation.)

Short Answer

Expert verified
Bob pays $7.50, Mary pays $15, Ivan pays $22.50, based on passenger-miles.

Step by step solution

01

Calculate the Total Distance

All together, from the starting point to the final stop, the taxi covers 30 miles. This is the distance from the starting point to Ivan's destination, which is the furthest.
02

Determine the Total Fare

Since the taxi costs $1.50 per mile, the total fare for the entire trip is calculated by multiplying the total distance by the cost per mile: \[ 1.50 \text{ dollars/mile} \times 30 \text{ miles} = 45 \text{ dollars} \]This is the total cost for the trip.
03

Assign Costs Based on Distance

Bob travels 10 miles, Mary travels 20 miles, and Ivan travels 30 miles. We can assign costs based on the distance each person travels. Bob pays \[ 1.50 \times 10 = 15 \text{ dollars} \]Mary pays \[ 1.50 \times 20 = 30 \text{ dollars} \]Ivan pays \[ 1.50 \times 30 = 45 \text{ dollars} \].
04

Consider Equal Share Approach

Another way to understand the fare split is to consider the total cost shared equally based on the total distance covered (30 miles total and 60 passenger-miles overall: 10 for Bob, 20 for Mary, and 30 for Ivan). The fare per passenger-mile is calculated as the total cost of 45 dollars divided by the total 60 passenger-miles:\[ \frac{45}{60} = 0.75 \text{ dollars per passenger-mile} \].Bob should pay \[ 0.75 \times 10 = 7.50 \text{ dollars} \],Mary pays \[ 0.75 \times 20 = 15 \text{ dollars} \],and Ivan pays \[ 0.75 \times 30 = 22.50 \text{ dollars} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cost Sharing
When people decide to share a cost, it means they are looking to divide an expense among themselves. This makes it affordable and fair for everyone involved. In the taxi problem, cost sharing allows Bob, Mary, and Ivan to split the ride cost. Together, they all benefit by paying less than they would have individually.

For this scenario, there are multiple ways to distribute the costs among passengers:
  • **By Distance: **Each pays according to the number of miles they travel.
  • **Equally: **Each shares an equal part of the total.
Both methods have their own reasoning, and understanding each thoroughly can help in more complex real-life situations too. Ultimately, cost sharing helps everyone involved to save money, while ensuring fair payment methods.
Proportional Reasoning
Proportional reasoning is a key mathematical concept that allows comparisons to be made rationally and fairly. It means understanding relationships between numbers based on multiplication and division.

In the taxi scenario, we see how the fare can be assigned differently by employing proportional reasoning. Using this approach involves:
  • **Calculating Total Costs: **Understand how each person's share relates to the total miles, like assigning costs per mile.
  • **Distance Relationships: **Realize that as each person travels a different distance, their share should reflect that.
By using proportional reasoning, Bob pays en route to his stop, while Mary and Ivan also pay for the further distances. It helps to ensure each payment is proportional to the distance they traveled. Essentially, this approach not only clarifies payment allocation using a distance-based strategy but also highlights equal sharing.
Problem-Solving Strategies
Effective problem-solving strategies are vital to tackle challenges in both math and daily life. They involve logical thinking and the ability to apply various strategies to find a solution. In this taxi fare exercise, these strategies can be seen in use:
  • **Identifying the Problem: **Determine that sharing costs equally or by distance is the issue.
  • **Exploring Options: **Looking at more than one way to calculate the shares for an appropriate solution.
  • **Solving Step by Step: **Breakdown of the problem into smaller parts to ensure understanding.
In the taxi problem, first, identify the total cost, then use different strategies for dividing it.
By honing such problem-solving abilities, you'll be equipped to handle similar issues with ease, not just in mathematics but in any collaborative financial effort.

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