Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To find the periodic payment that will amount to a future value of [tex]$21,000 when an interest rate of 6% (compounded annually) is applied over 12 consecutive years, we can proceed as follows:
### Step-by-Step Solution:
1. Understand the Terms and the Formula:
- Future Value (FV) = $[/tex]21,000
- Interest Rate (r) = 6% or 0.06 (as a decimal)
- Number of Periods (n) = 12 years
We will use the formula for the future value of an ordinary annuity:
[tex]\[ FV = PMT \times \left[\frac{{(1 + r)^n - 1}}{r}\right] \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value.
- [tex]\( PMT \)[/tex] is the periodic payment.
- [tex]\( r \)[/tex] is the interest rate per period.
- [tex]\( n \)[/tex] is the number of periods.
2. Rearrange the Formula to Solve for [tex]\( PMT \)[/tex]:
We need to solve for the periodic payment [tex]\( PMT \)[/tex]:
[tex]\[ PMT = \frac{FV}{\left[\frac{{(1 + r)^n - 1}}{r}\right]} \][/tex]
3. Substitute the Known Values:
- [tex]\( FV = 21000 \)[/tex]
- [tex]\( r = 0.06 \)[/tex]
- [tex]\( n = 12 \)[/tex]
So, the rearranged formula becomes:
[tex]\[ PMT = \frac{21000}{\left[\frac{{(1 + 0.06)^{12} - 1}}{0.06}\right]} \][/tex]
4. Calculate the Denominator:
Let's break this down:
- Calculate [tex]\( (1 + 0.06)^{12} \)[/tex]:
[tex]\[ (1 + 0.06)^{12} = 1.06^{12} \][/tex]
- Subtract 1 from this result:
[tex]\[ 1.06^{12} - 1 \][/tex]
- Divide by the interest rate [tex]\( r = 0.06 \)[/tex]:
[tex]\[ \frac{1.06^{12} - 1}{0.06} \][/tex]
5. Complete the Calculation:
After calculating the entire expression in the denominator, plug back into the formula:
[tex]\[ PMT = \frac{21000}{\left[\frac{(1 + 0.06)^{12} - 1}{0.06}\right]} \][/tex]
6. Round the Result:
Ensure the final answer is rounded to the nearest cent.
The periodic payment will amount to approximately $1244.82.
- Interest Rate (r) = 6% or 0.06 (as a decimal)
- Number of Periods (n) = 12 years
We will use the formula for the future value of an ordinary annuity:
[tex]\[ FV = PMT \times \left[\frac{{(1 + r)^n - 1}}{r}\right] \][/tex]
Where:
- [tex]\( FV \)[/tex] is the future value.
- [tex]\( PMT \)[/tex] is the periodic payment.
- [tex]\( r \)[/tex] is the interest rate per period.
- [tex]\( n \)[/tex] is the number of periods.
2. Rearrange the Formula to Solve for [tex]\( PMT \)[/tex]:
We need to solve for the periodic payment [tex]\( PMT \)[/tex]:
[tex]\[ PMT = \frac{FV}{\left[\frac{{(1 + r)^n - 1}}{r}\right]} \][/tex]
3. Substitute the Known Values:
- [tex]\( FV = 21000 \)[/tex]
- [tex]\( r = 0.06 \)[/tex]
- [tex]\( n = 12 \)[/tex]
So, the rearranged formula becomes:
[tex]\[ PMT = \frac{21000}{\left[\frac{{(1 + 0.06)^{12} - 1}}{0.06}\right]} \][/tex]
4. Calculate the Denominator:
Let's break this down:
- Calculate [tex]\( (1 + 0.06)^{12} \)[/tex]:
[tex]\[ (1 + 0.06)^{12} = 1.06^{12} \][/tex]
- Subtract 1 from this result:
[tex]\[ 1.06^{12} - 1 \][/tex]
- Divide by the interest rate [tex]\( r = 0.06 \)[/tex]:
[tex]\[ \frac{1.06^{12} - 1}{0.06} \][/tex]
5. Complete the Calculation:
After calculating the entire expression in the denominator, plug back into the formula:
[tex]\[ PMT = \frac{21000}{\left[\frac{(1 + 0.06)^{12} - 1}{0.06}\right]} \][/tex]
6. Round the Result:
Ensure the final answer is rounded to the nearest cent.
The periodic payment will amount to approximately $1244.82.
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
