Questions: Dante is going to invest to help with a down payment on a home. How much would he have to invest to have 46,900 after 7 years, assuming an interest rate of 1.65% compounded annually? Do not round any intermediate computations, and round your final answer to the nearest dollar. If necessary, refer to the list of financial formulas.

Solution Steps

To determine how much Dante needs to invest now to have $46,900 after 7 years with an interest rate of 1.65% compounded annually, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (the initial amount of money).
• \( r \) is the annual interest rate (decimal).
• \( n \) is the number of times that interest is compounded per year.
• \( t \) is the time the money is invested for in years.

We need to solve for \( P \):

\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Given:

• \( A = 46900 \)
• \( r = 0.0165 \)
• \( n = 1 \) (compounded annually)
• \( t = 7 \)

We are given:

• \( A = 46900 \)
• \( r = 0.0165 \)
• \( n = 1 \) (compounded annually)
• \( t = 7 \)

The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

We need to solve for \( P \): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Substitute the given values into the formula: \[ P = \frac{46900}{\left(1 + \frac{0.0165}{1}\right)^{1 \cdot 7}} \]

Simplify the expression inside the parentheses: \[ P = \frac{46900}{\left(1 + 0.0165\right)^7} \] \[ P = \frac{46900}{(1.0165)^7} \]

Calculate the value of \((1.0165)^7\): \[ (1.0165)^7 \approx 1.1209 \]

Now, divide the future value by the calculated denominator: \[ P = \frac{46900}{1.1209} \approx 41823.5759 \]

Round the principal amount to the nearest dollar: \[ P \approx 41824 \]

Final Answer