Questions: Tim wants to purchase a 25000 car in 5 years. How much money should he deposit each month into a bank account paying interest at the rate of 5% / year compounded monthly?

Tim wants to purchase a 25000 car in 5 years. How much money should he deposit each month into a bank account paying interest at the rate of 5% / year compounded monthly?

Transcript text: Tim wants to purchase a $\$ 25000$ car in 5 years. How much money should he deposit each month into a bank account paying interest at the rate of $5 \% /$ year compounded monthly?

Solution

Solution Steps

To solve this problem, we need to use the formula for the future value of a series of equal monthly deposits (an annuity). The future value of an annuity formula is used to determine how much Tim needs to deposit each month to reach his goal of $25,000 in 5 years, given a 5% annual interest rate compounded monthly. We will rearrange the formula to solve for the monthly deposit amount.

Step 1: Given Values

We are given the following values:

Future value ($ FV $) = 25000
Annual interest rate ($ r $) = 0.05
Compounding periods per year ($ n $) = 12
Number of years ($ t $) = 5

Step 2: Calculate Monthly Interest Rate

The monthly interest rate ($ i $) is calculated as: \[ i = \frac{r}{n} = \frac{0.05}{12} \approx 0.0041667 \]

Step 3: Calculate Total Number of Deposits

The total number of deposits ($ N $) over 5 years is: \[ N = n \times t = 12 \times 5 = 60 \]

Step 4: Calculate Monthly Deposit

Using the future value of an annuity formula, the monthly deposit ($ PMT $) can be calculated as: \[ PMT = \frac{FV \cdot i}{(1 + i)^N - 1} \] Substituting the known values: \[ PMT = \frac{25000 \cdot 0.0041667}{(1 + 0.0041667)^{60} - 1} \approx 367.6142 \]