Mark wants to buy a car in 15 months, when he graduates. He estimates the car he wants will cost $12 500. Mark has just invested $7500 in a GIC earning 4% compounded quarterly. He also has a savings account that earns 2.45%, compounded monthly. How much should he deposit in his savings account each month to have enough money to buy the car?
To calculate the amount Mark should deposit in his savings account each month, we need to consider the time frame, interest rates, and the total cost of the car.
First, let's calculate the future value of Mark's GIC investment after 15 months. We can use the compound interest formula:
A = P * (1 + r/n)^(nt)
Where:
A is the future value of the investment
P is the initial principal amount
r is the annual interest rate (expressed as a decimal)
n is the number of compounding periods per year
t is the number of years
Given:
P = $7500
r = 4% (or 0.04)
n = 4 (compounded quarterly)
t = 15/12 (converted to years)
A = 7500 * (1 + 0.04/4)^(4 * 15/12)
Calculating this, we find that the future value of Mark's GIC investment after 15 months is approximately $8019.12.
Next, let's consider Mark's savings account. We need to calculate the monthly deposit required to reach the remaining cost of the car after accounting for the GIC investment.
The remaining cost of the car is $12,500 - $8019.12 = $4680.88
Now, we can use the future value formula for a savings account with monthly deposits:
A = P * (1 + r)^t + D * ((1 + r)^t - 1) / r
Where:
A is the future value of the investment
P is the initial principal amount
r is the monthly interest rate (annual interest rate divided by 12)
t is the number of months
D is the monthly deposit amount
Given:
A = $4680.88
P = $0 (initial principal)
r = 2.45% (or 0.0245/12)
t = 15 months
$4680.88 = 0 * (1 + 0.0245/12)^15 + D * ((1 + 0.0245/12)^15 - 1) / (0.0245/12)
Simplifying the formula, we find:
4680.88 = D * ((1.0245)^15 - 1) / (0.0245/12)
Now, we solve for D:
D = 4680.88 * (0.0245/12) / ((1.0245)^15 - 1)
Evaluating this expression, we find that Mark should deposit approximately $237.71 in his savings account each month to have enough money to buy the car.
Therefore, Mark should deposit $237.71 into his savings account each month to reach his goal of buying the car in 15 months.