To find the interest earned on a principal of £2000 at a rate of 5% per annum for two years, compounded annually, follow these steps:
1. Understand the Given Values:
- Principal (P): £2000
- Annual Interest Rate (r): 5% (which is 0.05 as a decimal)
- Time Period (t): 2 years
- Number of times interest is compounded per year (n): 1 (since it is compounded annually)
2. Use the Compound Interest Formula:
- The formula for compound interest is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
- Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( n \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
3. Substitute the Given Values into the Formula:
- Substitute [tex]\( P = 2000 \)[/tex], [tex]\( r = 0.05 \)[/tex], [tex]\( n = 1 \)[/tex], and [tex]\( t = 2 \)[/tex] into the formula:
[tex]\[
A = 2000 \left(1 + \frac{0.05}{1}\right)^{1 \times 2}
\][/tex]
[tex]\[
A = 2000 \left(1 + 0.05\right)^2
\][/tex]
[tex]\[
A = 2000 \left(1.05\right)^2
\][/tex]
4. Calculate the Accumulated Amount:
- Calculate [tex]\( 1.05^2 \)[/tex]:
[tex]\[
1.05^2 = 1.1025
\][/tex]
- Multiply this by the principal:
[tex]\[
A = 2000 \times 1.1025 = 2205
\][/tex]
5. Determine the Interest Earned:
- The interest earned is the difference between the accumulated amount and the principal:
[tex]\[
\text{Interest Earned} = A - P
\][/tex]
[tex]\[
\text{Interest Earned} = 2205 - 2000 = 205
\][/tex]
Thus, the interest earned on a principal of £2000 at a rate of 5% per annum for two years, compounded annually, is £205.
