Calculating Bicycle Rental Costs A Step By Step Guide

Hiring a bicycle can be a convenient and enjoyable way to explore a new area or simply get some exercise. However, it's essential to understand the costs involved before making a decision. This article delves into the cost structure of hiring a bicycle, specifically focusing on a scenario with an upfront fee and a daily rental rate. We will explore how to formulate an equation to represent the total cost and utilize this equation to calculate the expenses for various rental durations. Understanding these calculations empowers you to budget effectively and make informed choices when considering bicycle rentals.

Understanding the Cost Structure

When renting a bicycle, the cost often involves a combination of an initial fee and a per-day charge. This structure is designed to cover the rental company's operational expenses and ensure profitability. The upfront cost typically accounts for the administrative tasks, bicycle maintenance, and initial preparation. The per-day charge reflects the usage of the bicycle and contributes to the overall revenue generated by the rental. By understanding this cost structure, you can better appreciate the pricing model and make informed decisions based on your needs and budget.

Problem Statement

The specific scenario we'll be examining involves a bicycle rental service with an upfront cost of £8 and a daily rental fee of £10. This means that regardless of how many days you rent the bicycle, you'll need to pay the initial £8. Subsequently, you'll be charged £10 for each day the bicycle is in your possession. Our goal is to develop an equation that accurately represents the total cost of hiring the bicycle for any given number of days. This equation will be a valuable tool for calculating rental expenses and making informed decisions about the rental duration.

Part A: Developing the Equation

Defining the Variables

To formulate the equation, we need to define the variables involved. Let's use 'c' to represent the total cost of hiring the bicycle in pounds, and 'd' to represent the number of days the bicycle is hired for. These variables will form the foundation of our equation, allowing us to express the relationship between the rental duration and the total expense. Carefully defining variables is crucial in mathematical modeling as it ensures clarity and facilitates accurate calculations.

Constructing the Equation

Now, let's construct the equation that represents the total cost. The upfront cost of £8 is a fixed amount, meaning it remains constant regardless of the rental duration. The daily cost of £10 is a variable amount, as it depends on the number of days the bicycle is hired. To calculate the total daily cost, we multiply the daily rate of £10 by the number of days, 'd'. Finally, we add the fixed upfront cost to the total daily cost to arrive at the total cost, 'c'. This logical breakdown allows us to translate the problem statement into a concise mathematical expression.

The equation can be written as follows:

c = 10d + 8

This equation encapsulates the entire cost structure. It states that the total cost ('c') is equal to 10 times the number of days ('d') plus the initial £8 fee. This equation is a powerful tool for predicting the cost of hiring the bicycle for any given number of days. By simply substituting the desired number of days for 'd', we can calculate the corresponding total cost 'c'.

Part B: Using the Equation

Creating a Table of Values

To illustrate the practical application of the equation, we will create a table of values. This table will show the total cost of hiring the bicycle for different durations. We will consider rental periods ranging from 1 to 7 days. This range provides a comprehensive overview of the cost progression and allows us to identify any patterns or trends. By examining the table, you can easily determine the cost associated with various rental durations and plan accordingly.

To complete the table, we will substitute each number of days (d) into the equation c = 10d + 8 and calculate the corresponding total cost (c). This process involves straightforward arithmetic operations and provides concrete examples of how the equation works in practice. The table will serve as a valuable reference for anyone considering renting the bicycle and wanting to estimate the expenses.

Calculations for Each Day

Let's perform the calculations for each day and populate the table:

• Day 1: c = 10(1) + 8 = 18
• Day 2: c = 10(2) + 8 = 28
• Day 3: c = 10(3) + 8 = 38
• Day 4: c = 10(4) + 8 = 48
• Day 5: c = 10(5) + 8 = 58
• Day 6: c = 10(6) + 8 = 68
• Day 7: c = 10(7) + 8 = 78

These calculations demonstrate how the total cost increases linearly with the number of days. For each additional day, the cost increases by £10, which is the daily rental rate. This linear relationship is a direct consequence of the equation we derived, where the daily cost is multiplied by the number of days.

Completed Table

This table clearly shows the total cost for renting the bicycle for each day from 1 to 7 days. You can quickly refer to this table to determine the rental expense for a specific duration. For example, if you plan to rent the bicycle for 3 days, the table indicates that the total cost will be £38. This visual representation of the data makes it easy to understand the cost implications of different rental durations.

Conclusion

In this article, we successfully formulated an equation to represent the cost of hiring a bicycle with an upfront fee and a daily rental rate. The equation, c = 10d + 8, accurately captures the relationship between the number of days the bicycle is hired for and the total cost. We then used this equation to create a table of values, which provides a clear and concise overview of the rental expenses for various durations. This process demonstrates the power of mathematical modeling in solving real-world problems and making informed decisions.

Understanding the cost structure of bicycle rentals, and indeed any service with a similar pricing model, is crucial for effective budgeting and financial planning. By breaking down the costs into fixed and variable components, we can develop equations that predict expenses and allow us to make informed choices. The skills and concepts explored in this article can be applied to a wide range of situations, from renting equipment to calculating service fees.

This exercise highlights the importance of mathematical literacy in everyday life. By understanding basic algebraic principles, we can analyze situations, develop models, and make informed decisions. Whether you're planning a cycling adventure or simply managing your finances, the ability to work with equations and interpret data is an invaluable asset.