The compound interest on a certain sum at a certain rate of interest for the second year and third year is ₹ 21780 and ₹ 23958, respectively. What is the rate of interest?

Options

A. `6%`
B. `12%`
C. `10%`
D. `15%`

Answer

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

### Step 1: Understand the relationship between the compound interests
The compound interest for the second year (CI2) and the third year (CI3) can be expressed in terms of the principal (P) and the rate of interest (r):
- CI2 = P * (r/100) * (1 + r/100)
- CI3 = P * (r/100) * (1 + r/100)^2

Given:
- CI2 = ₹ 21,780
- CI3 = ₹ 23,958

### Step 2: Set up the equations
From the information above, we can set up the following equations:
1. CI2 = P * (r/100) * (1 + r/100) = 21,780
2. CI3 = P * (r/100) * (1 + r/100)^2 = 23,958

### Step 3: Find the difference between CI3 and CI2
The difference between the compound interest for the third year and the second year gives us the interest on the amount at the end of the second year:
- CI3 - CI2 = P * (r/100)^2

Calculating the difference:
- 23,958 - 21,780 = 2,178
- So, P * (r/100)^2 = 2,178

### Step 4: Substitute CI2 into the equation
From the first equation, we can express P in terms of CI2:
- P = 21,780 / [(r/100) * (1 + r/100)]

### Step 5: Substitute P into the difference equation
Substituting P into the difference equation:
- (21,780 / [(r/100) * (1 + r/100)]) * (r/100)^2 = 2,178

### Step 6: Simplify the equation
This simplifies to:
- 21,780 * (r/100) = 2,178 * (1 + r/100)

### Step 7: Solve for r
Expanding and rearranging:
- 21,780 * (r/100) = 2,178 + 2,178 * (r/100)
- 21,780 * (r/100) - 2,178 * (r/100) = 2,178
- (21,780 - 2,178) * (r/100) = 2,178
- 19,602 * (r/100) = 2,178
- r/100 = 2,178 / 19,602
- r = (2,178 / 19,602) * 100

Calculating r:
- r ≈ 11.09%

### Final Answer
The rate of interest is approximately **11.09%**.

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