Question
Leo invests $2,000 at an interest rate of 4%, compounded quarterly, and another $2,000 at an interest rate of 3.75%, compounded annually. How much are the investments worth in total at the end of 3 years?
Answers
GPT-4o mini
To calculate the future value of Leo's investments, we will use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \( P \) = principal amount (the initial amount of money)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years
- \( A \) = the amount of money accumulated after \( n \) years, including interest.
### Investment 1: $2,000 at 4% compounded quarterly
- Principal \( P = 2000 \)
- Annual interest rate \( r = 4\% = 0.04 \)
- Compounded quarterly \( n = 4 \)
- Time \( t = 3 \) years
Substituting these values into the formula:
\[
A_1 = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 3}
\]
Calculating step-by-step:
1. Calculate \( \frac{0.04}{4} = 0.01 \)
2. Thus, \( A_1 = 2000 \left(1 + 0.01\right)^{12} \)
3. \( 1 + 0.01 = 1.01 \)
4. Calculate \( 1.01^{12} \):
\[
1.01^{12} \approx 1.126825
\]
5. Now, calculate \( A_1 \):
\[
A_1 = 2000 \times 1.126825 \approx 2253.65
\]
### Investment 2: $2,000 at 3.75% compounded annually
- Principal \( P = 2000 \)
- Annual interest rate \( r = 3.75\% = 0.0375 \)
- Compounded annually \( n = 1 \)
- Time \( t = 3 \) years
Substituting these values into the formula:
\[
A_2 = 2000 \left(1 + \frac{0.0375}{1}\right)^{1 \times 3}
\]
Calculating step-by-step:
1. Thus, \( A_2 = 2000 \left(1 + 0.0375\right)^{3} \)
2. \( 1 + 0.0375 = 1.0375 \)
3. Calculate \( 1.0375^{3} \):
\[
1.0375^{3} \approx 1.115267
\]
4. Now, calculate \( A_2 \):
\[
A_2 = 2000 \times 1.115267 \approx 2230.53
\]
### Total Value of Investments
Now, we add the values of both investments:
\[
\text{Total} = A_1 + A_2
\]
\[
\text{Total} \approx 2253.65 + 2230.53 \approx 4484.18
\]
Thus, the total worth of Leo's investments at the end of 3 years is approximately **$4,484.18**.
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \( P \) = principal amount (the initial amount of money)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = number of years
- \( A \) = the amount of money accumulated after \( n \) years, including interest.
### Investment 1: $2,000 at 4% compounded quarterly
- Principal \( P = 2000 \)
- Annual interest rate \( r = 4\% = 0.04 \)
- Compounded quarterly \( n = 4 \)
- Time \( t = 3 \) years
Substituting these values into the formula:
\[
A_1 = 2000 \left(1 + \frac{0.04}{4}\right)^{4 \times 3}
\]
Calculating step-by-step:
1. Calculate \( \frac{0.04}{4} = 0.01 \)
2. Thus, \( A_1 = 2000 \left(1 + 0.01\right)^{12} \)
3. \( 1 + 0.01 = 1.01 \)
4. Calculate \( 1.01^{12} \):
\[
1.01^{12} \approx 1.126825
\]
5. Now, calculate \( A_1 \):
\[
A_1 = 2000 \times 1.126825 \approx 2253.65
\]
### Investment 2: $2,000 at 3.75% compounded annually
- Principal \( P = 2000 \)
- Annual interest rate \( r = 3.75\% = 0.0375 \)
- Compounded annually \( n = 1 \)
- Time \( t = 3 \) years
Substituting these values into the formula:
\[
A_2 = 2000 \left(1 + \frac{0.0375}{1}\right)^{1 \times 3}
\]
Calculating step-by-step:
1. Thus, \( A_2 = 2000 \left(1 + 0.0375\right)^{3} \)
2. \( 1 + 0.0375 = 1.0375 \)
3. Calculate \( 1.0375^{3} \):
\[
1.0375^{3} \approx 1.115267
\]
4. Now, calculate \( A_2 \):
\[
A_2 = 2000 \times 1.115267 \approx 2230.53
\]
### Total Value of Investments
Now, we add the values of both investments:
\[
\text{Total} = A_1 + A_2
\]
\[
\text{Total} \approx 2253.65 + 2230.53 \approx 4484.18
\]
Thus, the total worth of Leo's investments at the end of 3 years is approximately **$4,484.18**.