A man borrowed 16,000 for 3 years under the following terms:
20% simple interest for the first 2 years.
20% C.I. for the remaining one year on the amount due after 2 years, the interest being compounded half-yearly.
Find the total amount to be paid at the end of three years.

Answer

To solve the problem step by step, we will first calculate the simple interest for the first 2 years, then determine the total amount after those 2 years, and finally calculate the compound interest for the last year, which is compounded half-yearly.

### Step 1: Calculate Simple Interest for the First 2 Years

**Formula for Simple Interest (SI):**
\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( P = 16,000 \) (Principal)
- \( R = 20\% \) (Rate of interest)
- \( T = 2 \) (Time in years)

**Calculation:**
\[ SI = \frac{16,000 \times 20 \times 2}{100} = \frac{16,000 \times 40}{100} = \frac{640,000}{100} = 6,400 \]

### Step 2: Calculate Total Amount After 2 Years

**Total Amount (A) after 2 years:**
\[ A = P + SI \]
\[ A = 16,000 + 6,400 = 22,400 \]

### Step 3: Calculate Compound Interest for the Last Year

For the last year, the interest is compounded half-yearly.

**New Principal (P) for Compound Interest:**
\[ P = 22,400 \]

**Rate and Time for Half-Yearly Compounding:**
- Rate for half-yearly compounding:
\[ R = \frac{20}{2} = 10\% \]
- Time for half-yearly compounding:
\[ T = 2 \text{ (because it is for 1 year compounded half-yearly, so 2 half-years)} \]

**Formula for Compound Interest (A):**
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

**Calculation:**
\[ A = 22,400 \left(1 + \frac{10}{100}\right)^2 = 22,400 \left(1 + 0.1\right)^2 = 22,400 \left(1.1\right)^2 \]
\[ A = 22,400 \times 1.21 = 27,104 \]

### Final Result

The total amount to be paid at the end of three years is:
\[ \text{Total Amount} = 27,104 \]

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