Buying A Car
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Module 1: Critical Thinking
Tony Spears
Feel free to handwrite any work and insert it as a picture.
Suppose you want to buy a new car and are trying to choose between two models:
Model A costs $16,500, its gas mileage is 25 miles per gallon, and insurance is $250 per year.
Model B costs $24,500, its gas mileage is 40 miles per gallon, and insurance is $450 per year.
If you drive approximately 40,000 miles per year and the gas costs $3 per gallon, answer the following questions:
1.
Find a formula for the total cost of owning Model A where the number of years you own the car is represented by x. Simplify your answer and present it in the form y = mx + b.
Formula for Model A: y = 14,650x + 16,500
Initial cost: $16,500
Annual gas cost: (40,000 miles / 25 mpg) * $3/gallon = 1,600 * 3 = $4,800
Annual insurance cost: $250
Thus, the total annual cost (excluding initial purchase) for Model A: $14,400 + $250 = $14,650
The formula, therefore, is: y = 14,650x+16,500
2.
Find a formula for the total cost of owning Model B where the number of years you own the car is represented by x. Simplify your answer and present it in the form y = mx + b.
Formula for Model B: y = 3,450x + 24,500
Initial cost: $24,500
Annual gas cost: (40,000 miles / 40 mpg) * $3/gallon = 1,000 * 3 = $3,000
Annual insurance cost: $450
Total annual cost for Model B: $3,000 + $450 = $3,450
Therefore, the formula for total cost for Model B: y = 3,450x + 24,500
3.
Find the total cost for each model for the first five years. This means finding the total cost for each model at the end of year 5. Show your work in the space below.
Model A cost after 5 years: $89,750 Model B cost after 5 years: $42,250
Model A: y = 14,650*5 + 16,500 = $89,750
Model B: y = 3,450*5 + 24,500 =$42,250
4.
If you plan to keep the car for 4 years, which model is more economical? What about if you plan to keep it for 6 years? Be sure to show the calculations for each vehicle at the end of year 4 and year 6, and put your answers in the spaces provided.
Model A cost after 4 years: $75,100 Model A cost after 6 years: $104,400
Model B cost after 4 years: $38,300 Model B cost after 6 years: 46,200
Which model is more economical: at 4 years: Model A or Model B at 6 years: Model A or Model B
At 4 years Model A: 14,650*4 + 16,500 = $75,100
Model B: 3,450*4 + 24,500 =$38,300
At 6 years:
Model A: 14,650*6 + 16,500 = $104,400
Model B: 3,450*6 + 24,500 = $46,200
5.
Find the number of years in which the total cost to keep the two cars will be the same.
Final Answer: Approximately 0.714 years
14,650x + 16,500 = 3,450x + 24,500 11,200x + 16,500 = 24,500
11,200x = 8,000 x = 8,000/11,200 = 5/7
Approximately 0.714 years
6.
Identify the number of months when neither car holds a cost of ownership advantage. This is similar to question 5, but it asks for months, not years.
Final Answer: 8.57 months.
5/7 * 12 months/year = 60/7 months = 8.57 months. 7.
What effect would the cost of gas doubling have on the cost of ownership? Be sure to show the calculations for each vehicle. The best way to answer this question is to modify your answer for questions 1 and 2 and present your answer in y=mx+b form.
Formula for Model A: y = 9,850x + 16,500
Formula for Model B: y = $6,450x + 24,500
Model A: Doubled Gas Cost = $6/gallon Annual Gas Cost with Doubled Price: (40,000 miles / 25 mpg) * $6 = 1,600 * 6 = $9,600
Total Cost: $9,600 + $250 = $9,850 The formula, therefore, is: y = 9,850x + 16,500 Model B
Annual Gas Cost with Doubled Price: (40,000 miles / 40 mpg) * $6 = 1,000 * 6 = $6,000
Total Cost: $6,000+$450 = $6,450 The formula becomes: y = $6,450x + 24,500 8.
If you can sell either car for 40% of its value at any time, how does the analysis change? Be sure to show the calculations for each vehicle. The best way to answer this question is to modify your answer for questions 1 and 2 and present your answer in y=mx+b form.
Formula for Model A:
y = 9,850x + 9,900
Formula for Model B: y = 6,450 + 14,700
Model A:
Resale Value: 16500 * 0.4 = 6600
Adjusted Initial Cost: 16500 - 6600 = 9900
Adjusted formula becomes: y = 9,850x + 9,900
Model B:
Resale Value: 24500 * 0.4 = 9800
Adjusted Initial Cost: 24500 - 9800 = 14700
The adjusted formula is: y = 6,450 + 14,700
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