A person invests Rs. 5,000 for two years at a certain rate of interest compound annually. At the end of one year, this sum amounts to Rs. 5,600. Calculate.
(i) the rate of interest per annum.
(ii) the amount at the end of the second year.

Answer

To solve the problem step by step, we will break it down into two parts: calculating the rate of interest per annum and then calculating the amount at the end of the second year.

### Step 1: Calculate the Rate of Interest

1. **Identify the given values:**
- Principal (P) = Rs. 5000
- Amount after 1 year (A) = Rs. 5600

2. **Use the formula for compound interest:**
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Here, \(n = 1\) (since we are calculating for one year).

3. **Substitute the known values into the formula:**
\[ 5600 = 5000 \left(1 + \frac{r}{100}\right)^1 \]

4. **Rearrange the equation to solve for \(r\):**
\[ 5600 = 5000 \left(1 + \frac{r}{100}\right) \]
Divide both sides by 5000:
\[ \frac{5600}{5000} = 1 + \frac{r}{100} \]

5. **Calculate \(\frac{5600}{5000}\):**
\[ \frac{5600}{5000} = 1.12 \]

6. **Now, isolate \(r\):**
\[ 1.12 - 1 = \frac{r}{100} \]
\[ 0.12 = \frac{r}{100} \]

7. **Multiply both sides by 100 to find \(r\):**
\[ r = 0.12 \times 100 = 12 \]

**Rate of Interest: \(r = 12\%\)**

### Step 2: Calculate the Amount at the End of the Second Year

1. **Use the compound interest formula again for two years:**
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Here, \(n = 2\).

2. **Substitute the known values:**
\[ A = 5000 \left(1 + \frac{12}{100}\right)^2 \]

3. **Calculate \(1 + \frac{12}{100}\):**
\[ 1 + 0.12 = 1.12 \]

4. **Now calculate \(A\):**
\[ A = 5000 \times (1.12)^2 \]

5. **Calculate \((1.12)^2\):**
\[ (1.12)^2 = 1.2544 \]

6. **Now substitute back to find \(A\):**
\[ A = 5000 \times 1.2544 = 6272 \]

**Amount at the end of the second year: \(A = Rs. 6272\)**

### Summary of Results
- (i) Rate of Interest per annum: **12%**
- (ii) Amount at the end of the second year: **Rs. 6272**