To determine the amount of interest earned by the mechanic after 3 years, we can use the compound interest formula. Here are the steps to find the interest:
1. Identify the Principal Amount (P):
The initial amount deposited by the mechanic is Sh 400,000.
2. Determine the Annual Interest Rate (r):
The annual interest rate is 10%, which is equivalent to 0.10 in decimal form.
3. Number of Years (t):
The amount of time the money is left to grow is 3 years.
4. Frequency of Compounding (n):
Since the interest is compounded annually, the frequency of compounding per year is 1.
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount,
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate in decimal,
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year,
- [tex]\( t \)[/tex] is the number of years.
5. Plug in the Values:
[tex]\[ A = 400,000 \left(1 + \frac{0.10}{1}\right)^{1 \times 3} \][/tex]
[tex]\[ A = 400,000 \left(1 + 0.10\right)^3 \][/tex]
[tex]\[ A = 400,000 \left(1.10\right)^3 \][/tex]
6. Calculate the Final Amount (A):
After performing the calculation, we find that the final amount [tex]\( A \)[/tex] is approximately Sh 532,400.00.
7. Determine the Interest Earned:
The interest earned can be found by subtracting the principal amount from the final amount:
[tex]\[ \text{Interest Earned} = A - P \][/tex]
[tex]\[ \text{Interest Earned} = 532,400.00 - 400,000 \][/tex]
[tex]\[ \text{Interest Earned} \approx 132,400.00 \][/tex]
Therefore, the mechanic's principal earned approximately Sh 132,400 in interest after 3 years.
