The compound interest on a sum for `5^(th)` year is ₹ 7800 and compound interest for sixth year is ₹ 9048 (interest is compounded annually). What is the rate of interest?

Options

A. `12%`
B. `15%`
C. `16%`
D. `14%`

Answer

To solve the problem of finding the rate of interest based on the given compound interest for the 5th and 6th years, we can follow these steps:

### Step 1: Understand the relationship between the compound interest for the 5th and 6th years.
The compound interest for the 5th year is given as ₹ 7800, and for the 6th year, it is ₹ 9048. The difference between the interest for these two years will help us find the rate of interest.

### Step 2: Calculate the difference in compound interest.
The difference in compound interest between the 6th year and the 5th year is:
\[ \text{Difference} = \text{CI for 6th year} - \text{CI for 5th year} = 9048 - 7800 = 1248 \]

### Step 3: Relate the difference to the principal and rate of interest.
The difference in compound interest between two consecutive years is equal to the principal amount at the beginning of the 6th year multiplied by the rate of interest (expressed as a decimal). Therefore, we can express this as:
\[ \text{Difference} = P \times \frac{R}{100} \]
Where \( P \) is the principal amount at the beginning of the 6th year, and \( R \) is the rate of interest.

### Step 4: Find the principal amount at the beginning of the 6th year.
The amount at the end of the 5th year is the principal for the 6th year. The amount at the end of the 5th year can be calculated as:
\[ \text{Amount at the end of 5th year} = \text{CI for 5th year} + \text{Principal} \]
Let’s denote the principal as \( P \). The amount at the end of the 5th year is:
\[ A_5 = P + 7800 \]
At the end of the 6th year, the amount is:
\[ A_6 = A_5 + 9048 = (P + 7800) + 9048 \]
Thus, the principal at the beginning of the 6th year is:
\[ P + 7800 \]

### Step 5: Set up the equation.
Substituting the principal into the difference equation, we have:
\[ 1248 = (P + 7800) \times \frac{R}{100} \]

### Step 6: Solve for the rate of interest.
We can express \( P + 7800 \) in terms of the amount at the end of the 5th year:
\[ P + 7800 = A_5 \]
Now, we need to express \( A_5 \) in terms of \( R \):
\[ A_5 = 7800 + P \]
Substituting this back into the equation gives us:
\[ 1248 = (A_5) \times \frac{R}{100} \]
To find \( R \), we can rearrange this equation:
\[ R = \frac{1248 \times 100}{A_5} \]

### Step 7: Calculate the value of \( R \).
Since we know \( A_5 \) is equal to \( P + 7800 \), we can assume \( P \) is the principal amount that we need to find. However, we can also find \( R \) directly from the difference in interest:
\[ R = \frac{1248 \times 100}{7800} \]
Calculating this gives:
\[ R = \frac{124800}{7800} = 16 \]

### Conclusion:
The rate of interest is \( R = 16\% \).