The compound interest compounded annually, on a certain sum is 29,040 in second year and is 31,944 in third year. Calculate.
the rate of interest.

Answer

To find the rate of interest given the compound interest amounts for the second and third years, we can follow these steps:

### Step 1: Identify the compound interest for the third year
The compound interest for the second year is given as 29,040, and for the third year, it is 31,944.

### Step 2: Calculate the interest earned in the third year
To find the interest earned in the third year, we subtract the compound interest of the second year from that of the third year:

\[ \text{Interest for third year} = \text{CI in third year} - \text{CI in second year} = 31,944 - 29,040 = 2,904 \]

### Step 3: Relate the interest to the principal and rate
The interest for the third year can also be expressed using the formula for simple interest:

\[ \text{Interest} = \frac{P \cdot r \cdot t}{100} \]

In this case, \( P \) is the amount at the end of the second year (which is the principal for the third year), \( r \) is the rate of interest we need to find, and \( t \) is 1 year.

### Step 4: Substitute the known values
From the previous steps, we know:
- Interest for the third year = 2,904
- Principal at the end of the second year = 29,040
- Time \( t = 1 \)

Substituting these values into the formula gives:

\[ 2,904 = \frac{29,040 \cdot r \cdot 1}{100} \]

### Step 5: Solve for \( r \)
To isolate \( r \), we can rearrange the equation:

\[ r = \frac{2,904 \cdot 100}{29,040} \]

### Step 6: Calculate \( r \)
Now we can perform the calculation:

\[ r = \frac{290,400}{29,040} = 10 \]

### Conclusion
Thus, the rate of interest is:

\[ \text{Rate of Interest} = 10\% \]

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