Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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How much must you deposit today into an account with quarterly compounding interest in an APR of 4.5% in order to have $25,000 in 8 years for down payment on a house? Assume no additional deposits are made.
![### Financial Mathematics Problem: Compound Interest Calculation
**Question:**
How much must you deposit today into an account with quarterly compounding interest and an APR of 4.5% in order to have $25,000 in 8 years for a down payment on a house? Assume no additional deposits are made.
#### Explanation:
To solve this problem, we need to use the formula for the present value of a compound interest account:
\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \( PV \) is the present value (the amount you need to deposit today),
- \( FV \) is the future value (the amount you want to have, $25,000),
- \( r \) is the annual interest rate (APR, 4.5% or 0.045),
- \( n \) is the number of times the interest is compounded per year (quarterly compounding, so \( n = 4 \)),
- \( t \) is the number of years the money is invested (8 years).
#### Calculation Steps:
1. Identify given values:
- \( FV = $25,000 \)
- \( r = 4.5\% = 0.045 \)
- \( n = 4 \)
- \( t = 8 \) years
2. Substitute the values into the formula:
\[ PV = \frac{25,000}{(1 + \frac{0.045}{4})^{4 \cdot 8}} \]
3. Simplify and solve the equation:
\[ PV = \frac{25,000}{(1 + 0.01125)^{32}} \]
\[ PV = \frac{25,000}{(1.01125)^{32}} \]
\[ PV = \frac{25,000}{1.4226516} \]
\[ PV \approx 17,579.55 \]
#### Conclusion:
In order to have $25,000 in 8 years for a down payment on a house, you must deposit approximately $17,579.55 into the account today, assuming quarterly compounding interest at an APR of 4.5% and no additional deposits.](https://content.bartleby.com/qna-images/question/c2527d55-bafb-4364-9044-dce9e5596181/7d234286-c9c3-44a2-ac9a-903f736ecdf3/h03wt5g_processed.jpeg)
Transcribed Image Text:### Financial Mathematics Problem: Compound Interest Calculation
**Question:**
How much must you deposit today into an account with quarterly compounding interest and an APR of 4.5% in order to have $25,000 in 8 years for a down payment on a house? Assume no additional deposits are made.
#### Explanation:
To solve this problem, we need to use the formula for the present value of a compound interest account:
\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \( PV \) is the present value (the amount you need to deposit today),
- \( FV \) is the future value (the amount you want to have, $25,000),
- \( r \) is the annual interest rate (APR, 4.5% or 0.045),
- \( n \) is the number of times the interest is compounded per year (quarterly compounding, so \( n = 4 \)),
- \( t \) is the number of years the money is invested (8 years).
#### Calculation Steps:
1. Identify given values:
- \( FV = $25,000 \)
- \( r = 4.5\% = 0.045 \)
- \( n = 4 \)
- \( t = 8 \) years
2. Substitute the values into the formula:
\[ PV = \frac{25,000}{(1 + \frac{0.045}{4})^{4 \cdot 8}} \]
3. Simplify and solve the equation:
\[ PV = \frac{25,000}{(1 + 0.01125)^{32}} \]
\[ PV = \frac{25,000}{(1.01125)^{32}} \]
\[ PV = \frac{25,000}{1.4226516} \]
\[ PV \approx 17,579.55 \]
#### Conclusion:
In order to have $25,000 in 8 years for a down payment on a house, you must deposit approximately $17,579.55 into the account today, assuming quarterly compounding interest at an APR of 4.5% and no additional deposits.
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