Acme Annuities recently offered an annuity that pays 5.4% compounded monthly. What equal monthly deposit should be made into this annuity in order to have $171,000 in 8 years? The amount of each deposit should be $ (Round to the nearest cent.)
Acme Annuities recently offered an annuity that pays 5.4% compounded monthly. What equal monthly deposit should be made into this annuity in order to have $171,000 in 8 years? The amount of each deposit should be $ (Round to the nearest cent.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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14
![**Annuity Calculation Problem**
**Question:**
Acme Annuities recently offered an annuity that pays 5.4% compounded monthly. What equal monthly deposit should be made into this annuity in order to have $171,000 in 8 years?
**Calculation:**
The amount of each deposit should be $____.
*Note: Round to the nearest cent.*
---
### Explanation:
To solve this problem, you need to determine the amount of an equal monthly deposit that will grow to a future value of $171,000 in 8 years, given the annuity pays 5.4% interest compounded monthly.
To approach this, you can use the future value of an annuity formula:
\[ A = P \frac{((1 + r)^n - 1)}{r} \]
Where:
- \( A \) is the future value of the annuity ($171,000).
- \( P \) is the monthly deposit.
- \( r \) is the monthly interest rate (annual rate divided by 12).
- \( n \) is the total number of deposits (years times 12).
Given:
\[ A = 171,000 \]
\[ Annual \: interest \: rate = 5.4\% \quad \Rightarrow \quad Monthly \: interest \: rate = \frac{5.4\%}{12} = 0.0045 \]
\[ n = 8 \, \text{years} \times 12 \, \text{months/year} = 96 \, \text{months} \]
Substitute these values into the formula and solve for \( P \).
This equation can be rearranged to solve for \( P \):
\[ P = \frac{A \cdot r}{((1 + r)^n - 1)} \]
Use a calculator to compute this value:
1. Calculate \( (1 + r)^n \).
2. Subtract 1 from the result.
3. Multiply the result by \( r \).
4. Divide \( A \) by the result.
Finally, round the answer to the nearest cent to find the monthly deposit amount.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d9df1eb-9cbe-4888-87d6-c14249115dd3%2F243943a2-3d8c-476f-bc5f-09aa47a84e48%2Fnt3fl44_processed.png&w=3840&q=75)
Transcribed Image Text:**Annuity Calculation Problem**
**Question:**
Acme Annuities recently offered an annuity that pays 5.4% compounded monthly. What equal monthly deposit should be made into this annuity in order to have $171,000 in 8 years?
**Calculation:**
The amount of each deposit should be $____.
*Note: Round to the nearest cent.*
---
### Explanation:
To solve this problem, you need to determine the amount of an equal monthly deposit that will grow to a future value of $171,000 in 8 years, given the annuity pays 5.4% interest compounded monthly.
To approach this, you can use the future value of an annuity formula:
\[ A = P \frac{((1 + r)^n - 1)}{r} \]
Where:
- \( A \) is the future value of the annuity ($171,000).
- \( P \) is the monthly deposit.
- \( r \) is the monthly interest rate (annual rate divided by 12).
- \( n \) is the total number of deposits (years times 12).
Given:
\[ A = 171,000 \]
\[ Annual \: interest \: rate = 5.4\% \quad \Rightarrow \quad Monthly \: interest \: rate = \frac{5.4\%}{12} = 0.0045 \]
\[ n = 8 \, \text{years} \times 12 \, \text{months/year} = 96 \, \text{months} \]
Substitute these values into the formula and solve for \( P \).
This equation can be rearranged to solve for \( P \):
\[ P = \frac{A \cdot r}{((1 + r)^n - 1)} \]
Use a calculator to compute this value:
1. Calculate \( (1 + r)^n \).
2. Subtract 1 from the result.
3. Multiply the result by \( r \).
4. Divide \( A \) by the result.
Finally, round the answer to the nearest cent to find the monthly deposit amount.
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