Sure, let's break down the solution for each part of the question step-by-step.
### Problem:
You invested [tex]$7000 in a bank account with an annual interest rate of 4%. You need to find the amount in the account after 11 years with different compounding periods: annually, quarterly, and monthly.
### Given:
- Principal amount (P): $[/tex]7000
- Annual interest rate (r): 4% or 0.04
- Time (t): 11 years
Let’s consider each case one by one.
### 1. Compounded Annually:
With annual compounding, interest is added once per year.
The formula for the compounded amount is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Since interest is compounded annually:
- [tex]\( n = 1 \)[/tex]
So, the formula simplifies to:
[tex]\[ A_{\text{annually}} = P \left(1 + r\right)^t \][/tex]
Plugging in the values:
[tex]\[ A_{\text{annually}} = 7000 \left(1 + 0.04\right)^{11} \][/tex]
After calculating, we get:
[tex]\[ A_{\text{annually}} = \$10776.18 \][/tex]
### 2. Compounded Quarterly:
With quarterly compounding, interest is added 4 times per year.
The formula remains the same, but with:
- [tex]\( n = 4 \)[/tex]
So:
[tex]\[ A_{\text{quarterly}} = P \left(1 + \frac{r}{4}\right)^{4t} \][/tex]
Plugging in the values:
[tex]\[ A_{\text{quarterly}} = 7000 \left(1 + \frac{0.04}{4}\right)^{4 \cdot 11} \][/tex]
After calculating, we get:
[tex]\[ A_{\text{quarterly}} = \$10845.22 \][/tex]
### 3. Compounded Monthly:
With monthly compounding, interest is added 12 times per year.
The formula remains the same, but with:
- [tex]\( n = 12 \)[/tex]
So:
[tex]\[ A_{\text{monthly}} = P \left(1 + \frac{r}{12}\right)^{12t} \][/tex]
Plugging in the values:
[tex]\[ A_{\text{monthly}} = 7000 \left(1 + \frac{0.04}{12}\right)^{12 \cdot 11} \][/tex]
After calculating, we get:
[tex]\[ A_{\text{monthly}} = \$10861.00 \][/tex]
### Summary:
- Amount after 11 years with annual compounding: \[tex]$10776.18
- Amount after 11 years with quarterly compounding: \$[/tex]10845.22
- Amount after 11 years with monthly compounding: \$10861.00
Make sure to double-check your calculations if you do it by hand or using a calculator to ensure accuracy.
