Question

You want to find out which fraction of your car's use is for commuting to work, and which is for personal use. You know the one-way distance from your home to work. For a particular period, you recorded the beginning and ending mileage on the odometer and the number of work days. Write an algorithm to settle this question.

Short answer

Calculate the total and work distances, subtract to find personal distance, then compute their fractions of total usage.

Step-by-step solution

Identify Problem Variables

Start by identifying and labeling all the variables involved in the problem. You have the one-way distance to work (let's call this \( d \)), the initial odometer reading at the start of the period (\( M_i \)), the final odometer reading at the end of the period (\( M_f \)), and the number of work days (\( W \)).

Calculate Total Distance Travelled

Calculate the total distance traveled during the period. You can find this by subtracting the initial odometer reading from the final odometer reading, i.e., \( \text{Total Distance} = M_f - M_i \).

Calculate Work-Related Distance

Calculate the total distance traveled to work and back. Since \( 2d \) is the round-trip distance for one workday, the total work-related distance is \( \text{Work Distance} = 2Wd \).

Calculate Personal Use Distance

Subtract the work-related distance from the total distance to determine the personal use distance, i.e., \( \text{Personal Distance} = \text{Total Distance} - \text{Work Distance} \).

Calculate Fractions of Use

Calculate the fraction of the car's use that was for commuting to work and for personal use. The fraction for work is \( \frac{\text{Work Distance}}{\text{Total Distance}} \) and for personal use is \( \frac{\text{Personal Distance}}{\text{Total Distance}} \).

Create and Execute Algorithm

Summarize the process into an algorithm: 1) Input \( d \), \( M_i \), \( M_f \), \( W \); 2) Calculate \( \text{Total Distance} = M_f - M_i \); 3) Calculate \( \text{Work Distance} = 2Wd \); 4) Calculate \( \text{Personal Distance} = \text{Total Distance} - \text{Work Distance} \); 5) Output \( \text{Work Fraction} = \frac{\text{Work Distance}}{\text{Total Distance}} \), \( \text{Personal Fraction} = \frac{\text{Personal Distance}}{\text{Total Distance}} \).

Key concepts

Variables Identification

When designing an algorithm, the first step is to correctly identify the variables you will work with. In this scenario, you are determining car usage for work versus personal purposes. To do this effectively, pinpoint the key data points matching these purposes:

• One-way distance to work (\( d \)): This is the distance from your home to your workplace.
• Initial odometer reading (\( M_i \)): Note the mileage of your car at the start of the period you're analyzing.
• Final odometer reading (\( M_f \)): Record the car's mileage at the end of the period.
• Number of workdays (\( W \)): Count how many days you actually traveled to work during the specified period.

Identifying these variables allows you to set a strong foundation for calculations, making sure you consider all necessary information for precise results.

Distance Calculation

Once you've identified all relevant variables, it's time to calculate the necessary distances. Begin with the total distance traveled during the period by your car. This figure is critical to decipher how much was for work versus personal use.
To find this, subtract the initial odometer reading (\( M_i \)) from the final odometer reading (\( M_f \)). The formula is:
\[ \text{Total Distance} = M_f - M_i \]
Next, discern the work-related distance. Since each workday involves a round trip from home to work and back, compute this by multiplying the number of workdays (\( W \)) by twice the one-way distance (\( 2d \)). Hence, the formula is:
\[ \text{Work Distance} = 2Wd \]
With these calculations, you can now delineate the portion of your driving that relates to commuting for work. This distance serves as a starting point for further analysis.

Fraction Calculation

The final step in understanding car usage is transforming the distance calculations into meaningful fractions of use. This will show exactly how much of the time your car is used for work and for personal purposes respectively.
Start by determining the personal distance, which is any mileage not accounted for by work-related trips. Do this by subtracting the work distance from the total distance:
\[ \text{Personal Distance} = \text{Total Distance} - \text{Work Distance} \]
Now, calculate the fractions:

• Work Fraction: This fraction indicates how much of the car's use was dedicated to work-related activities:
  \[ \text{Work Fraction} = \frac{\text{Work Distance}}{\text{Total Distance}} \]
• Personal Fraction: Similarly, this represents the portion of car use for personal reasons:
  \[ \text{Personal Fraction} = \frac{\text{Personal Distance}}{\text{Total Distance}} \]

In conclusion, these fractions provide clarity on vehicle usage patterns over the chosen period, and help in making informed decisions about your driving habits.

Odometer Reading Analysis

Analyzing odometer readings is a practical way to estimate driving habits over time. The odometer reflects the car's mileage, making it a critical tool in determining total distance traveled.
By subtracting the initial reading (\( M_i \)) from the final reading (\( M_f \)), the "Total Distance" driven is simply and accurately calculated.
Additionally, these measurements help distinguish between personal and work-related driving. This analysis illuminates not only how the car is used, but also how to potentially adjust use for better fuel efficiency or reduction in wear and tear.
Understanding odometer readings can thus aid in assessing your car's performance and lifespan, as well as planning for repairs and maintenance. It grants a quantifiable measure for optimizing automotive use, grounded in concrete travel data.