Problem 39 How much money do you have to pu... [FREE SOLUTION]

Chapter 17: Problem 39

How much money do you have to put into a bank account that pays $10 \%$ interest compounded annually to have $\$ 10,000$ in ten years?

Short Answer

Expert verified

To have $\$ 10,000$ in ten years with a 10% annual interest rate compounded annually, you need to initially deposit approximately $\$ 3855.43$ into the bank account.

Step by step solution

Understand the compound interest formula

The compound interest formula is given by: $A = P (1 + \frac{r}{n})^{nt}$, where: A - the final balance after t years P - the initial principal (amount of money deposited) r - the annual interest rate in decimal n - the number of times interest is compounded per year t - number of years In this exercise, we have: A = $10,000 r = 10% = 0.1 (converted to decimal) n = 1 (as the interest is compounded annually) t = 10 years Our goal is to find the initial principal (P) that satisfies this equation.

Rearrange the formula to solve for P

We need to find the value of P, so we will rearrange the formula as: $P = \frac{A}{(1 + \frac{r}{n})^{nt}}$ Now, we will plug in the values we have into the formula.

Substitute the values and solve

Substitute the values into the formula to find P: $P = \frac{10000}{(1 + \frac{0.1}{1})^{1*10}}$ Simplify the expression: $P = \frac{10000}{(1 + 0.1)^{10}}$ $P = \frac{10000}{(1.1)^{10}}$ Now, calculate the value of P: $P 鈮� 3855.43$

Interpret the result

The result shows that an initial deposit of about $ \ \$3855.43$ is needed to have $\$10,000$ in ten years with a 10% interest rate compounded annually.