Question

1. you would like to have $45,000 available after 10 years from now to start your own business. how much money do you need to save right now in the bank with an interest rate of 8.25% (compounding annually)? (4 points)

Explanation:

Step1: Recall compound interest formula for present value

The formula for compound interest when finding the present value \( P \) (the amount to save now) given the future value \( F \), annual interest rate \( r \), number of compounding periods per year \( n \), and number of years \( t \) is:
\( P = \frac{F}{(1 + \frac{r}{n})^{nt}} \)
Here, compounding is annual, so \( n = 1 \), \( F = 45000 \), \( r = 0.0825 \) (8.25% as a decimal), and \( t = 10 \).

Step2: Substitute values into the formula

Substitute \( F = 45000 \), \( r = 0.0825 \), \( n = 1 \), and \( t = 10 \) into the formula:
\( P = \frac{45000}{(1 + \frac{0.0825}{1})^{1 \times 10}} \)
Simplify the denominator:
\( (1 + 0.0825)^{10} = (1.0825)^{10} \)

Step3: Calculate \( (1.0825)^{10} \)

Using a calculator, \( (1.0825)^{10} \approx 2.20804 \)

Step4: Solve for \( P \)

Divide \( 45000 \) by \( 2.20804 \):
\( P \approx \frac{45000}{2.20804} \approx 20380.07 \)

Answer:

You need to save approximately \(\$20,380.07\) right now.