To solve the problem, we need to follow these steps:
a. Calculate Simple Interest
Simple interest can be calculated using the formula:
\[ \text{Simple Interest} = P \times r \times t \]
Where:
- \( P \) = Principal amount (initial investment)
- \( r \) = Annual interest rate (in decimal)
- \( t \) = Time (in years)
Given:
- \( P = K15000 \)
- \( r = 6% = 0.06 \)
- \( t = 30 \text{ months} = \frac{30}{12} \text{ years} = 2.5 \text{ years} \)
Now, plug in the values:
\[ \text{Simple Interest} = 15000 \times 0.06 \times 2.5 \]
Calculating this step-by-step:
- Calculate \( 0.06 \times 2.5 = 0.15 \)
- Calculate \( 15000 \times 0.15 = 2250 \)
Thus, the simple interest is: \[ \text{Simple Interest} = K2250 \]
b. Calculate Compound Interest Compounded Every Two Months
To calculate the compound interest, we can use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = Amount of money accumulated after n years, including interest.
- \( P \) = Principal amount (initial investment)
- \( r \) = Annual interest rate (in decimal)
- \( n \) = Number of times that interest is compounded per year
- \( t \) = Time (in years)
Given:
- \( P = K15000 \)
- \( r = 6% = 0.06 \)
- Compounding every two months means \( n = 6 \) (as there are 6 two-month periods in a year)
- \( t = 30 \text{ months} = \frac{30}{12} \text{ years} = 2.5 \text{ years} \)
Now, plug in the values:
\[ A = 15000 \left(1 + \frac{0.06}{6}\right)^{6 \times 2.5} \]
Calculate \( \frac{0.06}{6} = 0.01 \) and \( 6 \times 2.5 = 15 \).
Thus, the formula becomes:
\[ A = 15000 \left(1 + 0.01\right)^{15} \]
Calculate:
\[ A = 15000 \left(1.01\right)^{15} \]
Now calculate \( (1.01)^{15} \):
\[ (1.01)^{15} \approx 1.16079 \quad (\text{you may use a calculator for precision}) \]
Now, plug this back into the amount formula:
\[ A \approx 15000 \times 1.16079 \approx 17411.85 \]
Finally, the compound interest earned is:
\[ \text{Compound Interest} = A - P = 17411.85 - 15000 = K2411.85 \]
Summary of Results
a. The simple interest earned is K2250.
b. The amount after compounding every two months is approximately K17411.85, and the compound interest earned is approximately K2411.85.