Answer :
Sure, let's go through the step-by-step process to find the present value that will grow to [tex]$28,000 if the interest is 8% compounded quarterly for 16 quarters.
### Step 1: Understand the Variables
1. Future Value (FV): This is the amount of money we want to have in the future, which is $[/tex]28,000.
2. Interest Rate (r): This is the annual interest rate, which is 8% or 0.08 as a decimal.
3. Number of Compounding Periods (n): Since the interest is compounded quarterly, and there are 4 quarters in a year, over 4 years, we have 16 quarters.
4. Compounding Frequency: The interest is compounded quarterly, so we need to adjust the interest rate accordingly.
### Step 2: Calculate the Periodic Interest Rate
Since the interest is compounded quarterly, we divide the annual interest rate by 4:
[tex]\[ \text{Periodic Interest Rate} = \frac{0.08}{4} = 0.02 \][/tex]
### Step 3: Use the Present Value Formula
The formula for present value (PV) for compound interest is:
[tex]\[ \text{PV} = \frac{\text{FV}}{(1 + \text{Periodic Interest Rate})^n} \][/tex]
Now plug in the values we know:
1. [tex]\( \text{FV} = 28000 \)[/tex]
2. [tex]\( r = 0.02 \)[/tex]
3. [tex]\( n = 16 \)[/tex]
### Step 4: Calculate the Present Value
[tex]\[ \text{PV} = \frac{28000}{(1 + 0.02)^{16}} \][/tex]
### Step 5: Evaluate the Expression
First, calculate the term inside the parentheses:
[tex]\[ 1 + 0.02 = 1.02 \][/tex]
Then raise it to the power of 16:
[tex]\[ 1.02^{16} \approx 1.3728 \][/tex] (approximately)
Now, divide the future value by this result:
[tex]\[ \text{PV} = \frac{28000}{1.3728} \][/tex]
[tex]\[ \text{PV} \approx 20396.48 \][/tex]
### Step 6: Conclusion
The present value that will grow to [tex]$28,000 under these conditions is approximately $[/tex]20,396.48, rounded to the nearest cent.
2. Interest Rate (r): This is the annual interest rate, which is 8% or 0.08 as a decimal.
3. Number of Compounding Periods (n): Since the interest is compounded quarterly, and there are 4 quarters in a year, over 4 years, we have 16 quarters.
4. Compounding Frequency: The interest is compounded quarterly, so we need to adjust the interest rate accordingly.
### Step 2: Calculate the Periodic Interest Rate
Since the interest is compounded quarterly, we divide the annual interest rate by 4:
[tex]\[ \text{Periodic Interest Rate} = \frac{0.08}{4} = 0.02 \][/tex]
### Step 3: Use the Present Value Formula
The formula for present value (PV) for compound interest is:
[tex]\[ \text{PV} = \frac{\text{FV}}{(1 + \text{Periodic Interest Rate})^n} \][/tex]
Now plug in the values we know:
1. [tex]\( \text{FV} = 28000 \)[/tex]
2. [tex]\( r = 0.02 \)[/tex]
3. [tex]\( n = 16 \)[/tex]
### Step 4: Calculate the Present Value
[tex]\[ \text{PV} = \frac{28000}{(1 + 0.02)^{16}} \][/tex]
### Step 5: Evaluate the Expression
First, calculate the term inside the parentheses:
[tex]\[ 1 + 0.02 = 1.02 \][/tex]
Then raise it to the power of 16:
[tex]\[ 1.02^{16} \approx 1.3728 \][/tex] (approximately)
Now, divide the future value by this result:
[tex]\[ \text{PV} = \frac{28000}{1.3728} \][/tex]
[tex]\[ \text{PV} \approx 20396.48 \][/tex]
### Step 6: Conclusion
The present value that will grow to [tex]$28,000 under these conditions is approximately $[/tex]20,396.48, rounded to the nearest cent.
