Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our 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.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.