Question

Estimate the number of automobile repair shops in the city you live in by considering its population, how often an automobile needs repairs, and how many cars each shop can service per day. Then look in the yellow pages of your phone directory to see how accurate your estimate is. By what percentage was your estimate off?

Short answer

The estimate was off by 16% from the actual number of repair shops.

Step-by-step solution

Gather Initial Data

Start by gathering basic data. Assume the city's population is 500,000 and about 0.5 cars per person. This gives us 250,000 cars in the city. Determine the frequency of repairs; assume each car is serviced once per year. Find out how many cars a shop can service daily; assume 10 cars per day.

Calculate Annual Repairs Needed

Calculate the total number of repairs needed annually by multiplying the number of cars by the frequency of repairs: 250,000 cars \( \times \) 1 repair per year = 250,000 repairs per year.

Calculate Shop Capacity

Determine how many repairs one shop can do in a year. If one shop services 10 cars per day, and it operates 300 days a year, then each shop can handle 3,000 repairs annually (10 cars \( \times \) 300 days).

Estimate Number of Shops

To find the total number of repair shops needed, divide the annual number of repairs by the capacity of one shop: 250,000 repairs / 3,000 repairs per shop \( \approx \) 83.3. So, approximately 84 shops are needed.

Validate with Real Data

After finding the number of repair shops in the yellow pages or similar sources, let's say we find 100 repair shops listed.

Calculate Error Percentage

Calculate how off the estimate was using the formula: \( \left| \frac{Estimated - Actual}{Actual} \right| \times 100\% \). With an estimate of 84 and an actual count of 100, the error is \( \left| \frac{84 - 100}{100} \right| \times 100\% = 16\% \).

Key concepts

Population Estimation

When estimating the number of repair shops in a city, the first step is to understand the population and its correlation with the number of vehicles. The population provides a foundational figure to determine potential car ownership. For instance, if a city has 500,000 people and it is assumed there are about 0.5 cars per person, this results in 250,000 cars in the city.

Often, the correlation of cars per person can vary based on factors such as the city's public transport system or geographic size. To refine estimations, consider:

• The city's car ownership trends
• Availability and efficiency of public transportation
• Local economic conditions affecting car ownership rates

These factors can help provide a more precise number of vehicles, which is crucial for further steps in the estimation process.

Service Frequency

Assessing how often cars require servicing is pivotal to our estimation method. In this case, we assume that each car requires one repair annually. However, this frequency can depend on several conditions such as:

• Age and condition of vehicles
• Driving habits influenced by urban or suburban settings
• Climate and environmental conditions affecting vehicle wear

To strengthen our estimation, you might analyze local data like service records from repair shops if accessible. Even though one repair per year per car is a standard estimate, ongoing maintenance and unforeseen repairs can alter this number.

Repair Shop Capacity

Understanding the capacity of repair shops to service cars is crucial. In this analysis, an assumed shop services 10 cars daily and operates 300 days annually. This results in a shop handling about 3,000 repairs each year.

Factors that can influence shop capacity include:

• Number of mechanics and their efficiency
• The types of repairs handled (quick fixes vs. extensive repairs)
• Tecnhological advancements affecting repair times

For a more accurate estimate, consider local shop data if available to understand actual workflow constraints and potential expansion capabilities in repair infrastructure.

Error Calculation

After estimating the number of repair shops based on population, service frequency, and shop capacity, verifying and adjusting your estimation is necessary. By comparing with actual data, such as listed shops in directories, you can calculate the approximation error.

In the given example, with a calculation of 84 needed shops but 100 found in reality, we use the formula: \[ \left| \frac{Estimated - Actual}{Actual} \right| \times 100\% \] This gives an error of 16%. This percentage illustrates the deviation from actual figures, where several factors might contribute, such as:

• Newly opened or closed shops not accounted for in quick estimates
• Regional differences in calculation assumptions
• Errors in initial population or car ownership assumptions

This feedback loop helps refine the initial model, making future estimates more accurate and adjusted for local dynamics.