Sure! Let's solve the problem step by step.
a) Calculate the total amount:
Given:
- Principal amount (P) = [tex]$10,000
- Annual interest rate (r) = 9% or 0.09 (as a decimal)
- Time (t) = 3 years
- Number of times interest is compounded per year (n) = 1 (since it is compounded annually)
The formula for calculating the total amount (A) using compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Plugging in the values:
\[ A = 10000 \left(1 + \frac{0.09}{1}\right)^{1 \times 3} \]
First, calculate the inside of the parentheses:
\[ 1 + \frac{0.09}{1} = 1.09 \]
Now raise this to the power of (n \times t):
\[ (1.09)^3 = 1.09 \times 1.09 \times 1.09 \]
After calculating, we get the total amount (A):
\[ A \approx 12950.29 \]
So, the total amount after 3 years is approximately $[/tex]12,950.29.
b) Calculate the compound interest:
Compound interest (CI) is the difference between the total amount (A) and the principal amount (P).
[tex]\[ CI = A - P \][/tex]
We already have the values for A and P:
[tex]\[ CI = 12950.29 - 10000 \][/tex]
[tex]\[ CI = 2950.29 \][/tex]
Therefore, the compound interest earned over 3 years is approximately [tex]$2,950.29.
Summary:
a) The total amount after 3 years is approximately $[/tex]12,950.29.
b) The compound interest earned over 3 years is approximately $2,950.29.
