John borrowed Rs 20,000 for 4 years under the following conditions :
10% simple interest for the first `2(1)/(2)` years.
10% C.I. for the remaining one and a half years on the amount due after `2(1)/(2)` years, the interest being compounded half-yearly.
Find the total amount to be paid at the end of fourth year.

Answer

To solve the problem step by step, we will first calculate the simple interest for the first 2.5 years and then calculate the compound interest for the remaining 1.5 years.

### Step 1: Calculate Simple Interest for the First 2.5 Years

The formula for Simple Interest (SI) is given by:

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount = Rs 20,000
- \( R \) = Rate of interest = 10%
- \( T \) = Time in years = 2.5 years

Substituting the values into the formula:

\[ SI = \frac{20000 \times 10 \times 2.5}{100} \]

Calculating the above:

\[ SI = \frac{20000 \times 10 \times 2.5}{100} = \frac{500000}{100} = 5000 \]

### Step 2: Calculate the Total Amount after 2.5 Years

To find the total amount after 2.5 years, we add the simple interest to the principal:

\[ \text{Total Amount} = P + SI = 20000 + 5000 = 25000 \]

### Step 3: Calculate Compound Interest for the Remaining 1.5 Years

Now, we will calculate the compound interest on the new principal amount of Rs 25,000 for the remaining 1.5 years. Since the interest is compounded half-yearly, we need to adjust the rate and time:

- The rate per half-year = \( \frac{10\%}{2} = 5\% \)
- The time in half-years = \( 1.5 \text{ years} = 3 \text{ half-years} \)

The formula for Compound Interest (CI) is given by:

\[ A = P \left(1 + \frac{R}{100}\right)^n \]

Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal amount = Rs 25,000
- \( R \) = Rate of interest per half-year = 5%
- \( n \) = Number of half-years = 3

Substituting the values into the formula:

\[ A = 25000 \left(1 + \frac{5}{100}\right)^3 \]

Calculating the above:

\[ A = 25000 \left(1 + 0.05\right)^3 = 25000 \left(1.05\right)^3 \]

Calculating \( (1.05)^3 \):

\[ (1.05)^3 = 1.157625 \]

Now substituting back:

\[ A = 25000 \times 1.157625 = 28940.625 \]

### Step 4: Final Amount to be Paid

The total amount to be paid at the end of the fourth year is approximately:

\[ \text{Total Amount} \approx 28940.63 \]

### Summary of the Solution

The total amount John needs to pay at the end of the fourth year is Rs 28,940.63.

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