Question
Chris invests $15,000 in a retirement account with an annual compound interest rate of 3.4%. Find the total amount of the investment after 27 years.
Round answers to the nearest cent (hundredth).
*Remember to use your STAAR Reference Sheet for support!
(1 point)
Responses
$13,770.00
$13,770.00
$28,770.00
$28,770.00
$36,995.24
$36,995.24
$21,995.24
When Martin was born, his father set up a $5,000 college fund for him with an annual compound interest rate of 7.3%.
What is the total value of this account after 18 years, and what was the interest earned on the account? Round the answers to two decimal places.
*Remember to use your STAAR Reference Sheet for support!
(2 points)
The final amount of the college fund is $
, and the amount of interest earned on the account is $
Round answers to the nearest cent (hundredth).
*Remember to use your STAAR Reference Sheet for support!
(1 point)
Responses
$13,770.00
$13,770.00
$28,770.00
$28,770.00
$36,995.24
$36,995.24
$21,995.24
When Martin was born, his father set up a $5,000 college fund for him with an annual compound interest rate of 7.3%.
What is the total value of this account after 18 years, and what was the interest earned on the account? Round the answers to two decimal places.
*Remember to use your STAAR Reference Sheet for support!
(2 points)
The final amount of the college fund is $
, and the amount of interest earned on the account is $
Answers
GPT-4o mini
To solve these problems, we will use the formula for compound interest:
\[
A = P(1 + r/n)^{nt}
\]
Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial investment).
- \(r\) = the annual interest rate (decimal).
- \(n\) = the number of times that interest is compounded per year.
- \(t\) = the number of years the money is invested for.
For both scenarios, we'll assume the interest is compounded annually, so \(n = 1\).
### First Scenario: Chris's Retirement Investment
**Given:**
- \(P = 15000\)
- \(r = 3.4\% = 0.034\)
- \(n = 1\)
- \(t = 27\)
**Calculation:**
\[
A = 15000(1 + 0.034/1)^{1 \cdot 27}
\]
\[
A = 15000(1 + 0.034)^{27}
\]
\[
A = 15000(1.034)^{27}
\]
Calculating \( (1.034)^{27} \):
\[
(1.034)^{27} \approx 2.417744
\]
Now plug this value back into the equation:
\[
A \approx 15000 \times 2.417744 \approx 36266.16
\]
So the total amount of the investment after 27 years is approximately **$36,266.16**. Rounding to the nearest cent gives us **$36,266.16**.
Since the closest answer in the provided options is **$36,995.24**, we'll choose that as it seems to be an estimation error in calculations or selections.
### Second Scenario: Martin's College Fund
**Given:**
- \(P = 5000\)
- \(r = 7.3\% = 0.073\)
- \(n = 1\)
- \(t = 18\)
**Calculation:**
\[
A = 5000(1 + 0.073/1)^{1 \cdot 18}
\]
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Calculating \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.61629
\]
Now plug this value back into the equation:
\[
A \approx 5000 \times 3.61629 \approx 18081.45
\]
So the final amount of the college fund is approximately **$18,081.45**.
### Interest Earned:
To find the interest earned, we subtract the principal from the total amount:
\[
\text{Interest} = A - P = 18081.45 - 5000 = 13081.45
\]
Therefore, the final values are as follows:
- The final amount of the college fund is **$18,081.45** and the amount of interest earned is **$13,081.45**.
\[
A = P(1 + r/n)^{nt}
\]
Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial investment).
- \(r\) = the annual interest rate (decimal).
- \(n\) = the number of times that interest is compounded per year.
- \(t\) = the number of years the money is invested for.
For both scenarios, we'll assume the interest is compounded annually, so \(n = 1\).
### First Scenario: Chris's Retirement Investment
**Given:**
- \(P = 15000\)
- \(r = 3.4\% = 0.034\)
- \(n = 1\)
- \(t = 27\)
**Calculation:**
\[
A = 15000(1 + 0.034/1)^{1 \cdot 27}
\]
\[
A = 15000(1 + 0.034)^{27}
\]
\[
A = 15000(1.034)^{27}
\]
Calculating \( (1.034)^{27} \):
\[
(1.034)^{27} \approx 2.417744
\]
Now plug this value back into the equation:
\[
A \approx 15000 \times 2.417744 \approx 36266.16
\]
So the total amount of the investment after 27 years is approximately **$36,266.16**. Rounding to the nearest cent gives us **$36,266.16**.
Since the closest answer in the provided options is **$36,995.24**, we'll choose that as it seems to be an estimation error in calculations or selections.
### Second Scenario: Martin's College Fund
**Given:**
- \(P = 5000\)
- \(r = 7.3\% = 0.073\)
- \(n = 1\)
- \(t = 18\)
**Calculation:**
\[
A = 5000(1 + 0.073/1)^{1 \cdot 18}
\]
\[
A = 5000(1 + 0.073)^{18}
\]
\[
A = 5000(1.073)^{18}
\]
Calculating \( (1.073)^{18} \):
\[
(1.073)^{18} \approx 3.61629
\]
Now plug this value back into the equation:
\[
A \approx 5000 \times 3.61629 \approx 18081.45
\]
So the final amount of the college fund is approximately **$18,081.45**.
### Interest Earned:
To find the interest earned, we subtract the principal from the total amount:
\[
\text{Interest} = A - P = 18081.45 - 5000 = 13081.45
\]
Therefore, the final values are as follows:
- The final amount of the college fund is **$18,081.45** and the amount of interest earned is **$13,081.45**.