Answer :
Certainly! Let's break down the steps to find the present value that will grow to [tex]$29,000, given an interest rate of 9% compounded quarterly for a total of 10 quarters.
### Step-by-Step Solution:
1. Identify the Given Parameters:
- Future Value (FV) = $[/tex]29,000
- Annual Interest Rate (r) = 9% or 0.09
- Number of Compounding Periods per Year (n) = 4 (quarterly)
- Total number of quarters (t) = 10
2. Calculate the Effective Interest Rate per Compounding Period:
The effective rate per period is the annual interest rate divided by the number of compounding periods per year.
[tex]\[ \text{Effective Interest Rate} = \frac{r}{n} = \frac{0.09}{4} = 0.0225 \][/tex]
3. Use the Present Value Formula for Compound Interest:
The present value (PV) formula for compound interest is:
[tex]\[ PV = \frac{FV}{(1 + \text{effective interest rate})^{t}} \][/tex]
Here, the effective interest rate per quarter is 0.0225 and the total number of quarters is 10.
4. Substitute the Values into the Formula:
[tex]\[ PV = \frac{29000}{(1 + 0.0225)^{10}} \][/tex]
Simplify the expression inside the parentheses first:
[tex]\[ 1 + 0.0225 = 1.0225 \][/tex]
Next, raise this to the power of 10:
[tex]\[ 1.0225^{10} \][/tex]
Using a calculator or appropriate computational tool, determine the value of [tex]\(1.0225^{10}\)[/tex]:
[tex]\[ 1.0225^{10} \approx 1.24874 \][/tex]
5. Divide the Future Value by this Result:
[tex]\[ PV = \frac{29000}{1.24874} \approx 23214.79 \][/tex]
6. Round to the Nearest Cent:
The precise present value, rounded to the nearest cent, is [tex]$23,214.79. ### Final Answer: The present value that will grow to $[/tex]29,000 if interest is 9% compounded quarterly for 10 quarters is $23,214.79.
- Annual Interest Rate (r) = 9% or 0.09
- Number of Compounding Periods per Year (n) = 4 (quarterly)
- Total number of quarters (t) = 10
2. Calculate the Effective Interest Rate per Compounding Period:
The effective rate per period is the annual interest rate divided by the number of compounding periods per year.
[tex]\[ \text{Effective Interest Rate} = \frac{r}{n} = \frac{0.09}{4} = 0.0225 \][/tex]
3. Use the Present Value Formula for Compound Interest:
The present value (PV) formula for compound interest is:
[tex]\[ PV = \frac{FV}{(1 + \text{effective interest rate})^{t}} \][/tex]
Here, the effective interest rate per quarter is 0.0225 and the total number of quarters is 10.
4. Substitute the Values into the Formula:
[tex]\[ PV = \frac{29000}{(1 + 0.0225)^{10}} \][/tex]
Simplify the expression inside the parentheses first:
[tex]\[ 1 + 0.0225 = 1.0225 \][/tex]
Next, raise this to the power of 10:
[tex]\[ 1.0225^{10} \][/tex]
Using a calculator or appropriate computational tool, determine the value of [tex]\(1.0225^{10}\)[/tex]:
[tex]\[ 1.0225^{10} \approx 1.24874 \][/tex]
5. Divide the Future Value by this Result:
[tex]\[ PV = \frac{29000}{1.24874} \approx 23214.79 \][/tex]
6. Round to the Nearest Cent:
The precise present value, rounded to the nearest cent, is [tex]$23,214.79. ### Final Answer: The present value that will grow to $[/tex]29,000 if interest is 9% compounded quarterly for 10 quarters is $23,214.79.
