Question

John invested K15000 at an interest rate of 6% p.a. for 30 months.
a. Calculate his simple interest
b. Calculate his installments compounded every two months.

Answer 1

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:

1. Calculate \( 0.06 \times 2.5 = 0.15 \)
2. 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**.