Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Grace's bank has a Cerificate of Deposit with an APR of 5% compounded monthly.
(a) If Grace deposits $1000 in the account, what will be the value in 5 years?
(b) If Grace needs $10,000 in 5 years, how much should she deposit monthly?
![**Certificate of Deposit Calculations**
Grace's bank has a Certificate of Deposit (CD) with an Annual Percentage Rate (APR) of 5%, compounded monthly.
**(a) Future Value Calculation:**
*Problem:* If Grace deposits $1000 in the account, what will be the value in 5 years?
To solve this, we use the future value formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) = the future value of the investment/loan, including interest.
- \( P \) = the principal investment amount (the initial deposit) = $1000.
- \( r \) = the annual interest rate (decimal) = 0.05.
- \( n \) = the number of times that interest is compounded per year = 12.
- \( t \) = the time the money is invested for in years = 5.
Plugging in the values, we get:
\[ A = 1000 \left(1 + \frac{0.05}{12}\right)^{12 \times 5} \]
**(b) Monthly Deposit Calculation:**
*Problem:* If Grace needs $10,000 in 5 years, how much should she deposit monthly?
To solve this, we use the future value of an annuity formula:
\[ A = P \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \]
where:
- \( A \) = the future value of the annuity (savings goal) = $10,000.
- \( P \) = the monthly deposit amount.
- \( r \) = the annual interest rate (decimal) = 0.05.
- \( n \) = the number of times the interest is compounded per year = 12.
- \( t \) = the number of years = 5.
Rearranging the formula to solve for \( P \), we get:
\[ P = \frac{A \times \frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1} \]
Substituting the known values:
\[ P = \frac{10000 \times \frac{0.05}{12}}{\left(1 + \frac{](https://content.bartleby.com/qna-images/question/c2527d55-bafb-4364-9044-dce9e5596181/9e93621c-81e5-4653-8b54-eb8d518c9ebf/6oign8p_processed.png)
Transcribed Image Text:**Certificate of Deposit Calculations**
Grace's bank has a Certificate of Deposit (CD) with an Annual Percentage Rate (APR) of 5%, compounded monthly.
**(a) Future Value Calculation:**
*Problem:* If Grace deposits $1000 in the account, what will be the value in 5 years?
To solve this, we use the future value formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) = the future value of the investment/loan, including interest.
- \( P \) = the principal investment amount (the initial deposit) = $1000.
- \( r \) = the annual interest rate (decimal) = 0.05.
- \( n \) = the number of times that interest is compounded per year = 12.
- \( t \) = the time the money is invested for in years = 5.
Plugging in the values, we get:
\[ A = 1000 \left(1 + \frac{0.05}{12}\right)^{12 \times 5} \]
**(b) Monthly Deposit Calculation:**
*Problem:* If Grace needs $10,000 in 5 years, how much should she deposit monthly?
To solve this, we use the future value of an annuity formula:
\[ A = P \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \]
where:
- \( A \) = the future value of the annuity (savings goal) = $10,000.
- \( P \) = the monthly deposit amount.
- \( r \) = the annual interest rate (decimal) = 0.05.
- \( n \) = the number of times the interest is compounded per year = 12.
- \( t \) = the number of years = 5.
Rearranging the formula to solve for \( P \), we get:
\[ P = \frac{A \times \frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1} \]
Substituting the known values:
\[ P = \frac{10000 \times \frac{0.05}{12}}{\left(1 + \frac{
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