As part of your financial planning, you wish to purchase a new cat exactly 5 years from today. The car you wish to purchase coasts $14,000, and your research indicates that its price will increase by 2% to 4% per year over the next 5 years.
a. Estimate the price of the cars at the end of 5 years if inflation is (1) 2% per year and (2) 4% per year.
b. How much more expensive will the car be if the rate of inflation is 4% rather than 2%?
To estimate the price of the car at the end of 5 years, we will use the formula for compound interest, since the price of the car is expected to increase by a certain percentage each year.
The formula for compound interest is:
A = P(1 + r)^n
Where:
A = Final amount (price of the car at the end of 5 years)
P = Initial amount (current price of the car)
r = Interest rate per year (as a decimal)
n = Number of years
Now let's calculate the answers to your questions:
a. Estimate the price of the car at the end of 5 years if inflation is:
1. 2% per year:
Given:
P = $14,000 (current price)
r = 2% or 0.02
n = 5 (number of years)
Plugging these values into the formula:
A = $14,000(1 + 0.02)^5
A ≈ $15,637.13
Therefore, if inflation is 2% per year, the estimated price of the car at the end of 5 years will be around $15,637.13.
2. 4% per year:
Given:
P = $14,000 (current price)
r = 4% or 0.04
n = 5 (number of years)
Plugging these values into the formula:
A = $14,000(1 + 0.04)^5
A ≈ $16,494.27
Therefore, if inflation is 4% per year, the estimated price of the car at the end of 5 years will be around $16,494.27.
b. To find out how much more expensive the car will be if the rate of inflation is 4% rather than 2%, we need to calculate the difference between the estimated prices in both cases.
Difference = Price at 4% - Price at 2%
Difference = $16,494.27 - $15,637.13
Difference ≈ $857.14
Therefore, the car will be approximately $857.14 more expensive if the rate of inflation is 4% rather than 2%.
To estimate the price of the car at the end of 5 years, we need to calculate the price increase for each year and add it to the initial price. Let's go step by step:
a. Estimating the price of the car at the end of 5 years:
1. Calculate the price increase for each year if inflation is 2% per year:
- Year 1: $14,000 * 2% = $280 increase
- Year 2: $14,000 * 2% = $280 increase
- Year 3: $14,000 * 2% = $280 increase
- Year 4: $14,000 * 2% = $280 increase
- Year 5: $14,000 * 2% = $280 increase
2. Add the price increases for all 5 years to the initial price:
$14,000 + $280 + $280 + $280 + $280 + $280 = $14,400
Therefore, if inflation is 2% per year, the estimated price of the car at the end of 5 years would be $14,400.
Now let's calculate the price with a 4% inflation rate per year:
1. Calculate the price increase for each year if inflation is 4% per year:
- Year 1: $14,000 * 4% = $560 increase
- Year 2: $14,000 * 4% = $560 increase
- Year 3: $14,000 * 4% = $560 increase
- Year 4: $14,000 * 4% = $560 increase
- Year 5: $14,000 * 4% = $560 increase
2. Add the price increases for all 5 years to the initial price:
$14,000 + $560 + $560 + $560 + $560 + $560 = $16,800
Therefore, if inflation is 4% per year, the estimated price of the car at the end of 5 years would be $16,800.
b. To find out how much more expensive the car will be if the rate of inflation is 4% rather than 2%, subtract the estimated price with 2% inflation from the estimated price with 4% inflation:
$16,800 - $14,400 = $2,400
Therefore, if the rate of inflation is 4% rather than 2%, the car will be $2,400 more expensive.