Certainly! Let's solve this step-by-step to find out what your bank balance will be if you put [tex]$4,000 in your account and earn a 5% interest rate compounded quarterly for six years.
### Step-by-Step Solution:
1. Identify the given values:
- Principal amount (\( P \)): $[/tex]4000
- Annual interest rate ([tex]\( r \)[/tex]): 5% or 0.05 as a decimal
- Number of times the interest is compounded per year ([tex]\( n \)[/tex]): 4 (quarterly)
- Time the money is invested for ([tex]\( t \)[/tex]): 6 years
2. Write down the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
- Where:
- [tex]\( A \)[/tex] = amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] = principal amount (the initial amount of money)
- [tex]\( r \)[/tex] = annual interest rate (decimal)
- [tex]\( n \)[/tex] = number of times the interest is compounded per year
- [tex]\( t \)[/tex] = time the money is invested for, in years
3. Plug in the given values:
[tex]\[
A = 4000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 6}
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
A = 4000 \left(1 + 0.0125\right)^{24}
\][/tex]
[tex]\[
A = 4000 \left(1.0125\right)^{24}
\][/tex]
5. Calculate the value of [tex]\( \left(1.0125\right)^{24} \)[/tex]:
[tex]\[
1.0125^{24} \approx 1.34735105041435
\][/tex]
6. Multiply this value by the principal amount ([tex]$4000):
\[
A = 4000 \times 1.34735105041435
\]
\[
A \approx 5389.4042016574
\]
### Final Result:
- The total amount of money accumulated after 6 years, including interest, is approximately $[/tex]5389.40.
- To find the interest earned, subtract the principal amount from the total amount:
[tex]\[
\text{Interest Earned} = A - P
\][/tex]
[tex]\[
\text{Interest Earned} = 5389.40 - 4000
\][/tex]
[tex]\[
\text{Interest Earned} \approx 1389.40
\][/tex]
Therefore, after 6 years, your bank balance will be approximately [tex]$5389.40, and the interest earned will be around $[/tex]1389.40.
