If C.I. of a certain sum at the end of 2 years and 3 years are ₹ 234 and ₹ 381 respectively. Find the rate of interest?

Options

A. `14.28%`
B. `18.33%`
C. `16.67%`
D. `12.5% `

Answer

To find the rate of interest given the compound interest (C.I.) for 2 years and 3 years, we can follow these steps:

### Step 1: Identify the given values
- C.I. for 2 years = ₹ 234
- C.I. for 3 years = ₹ 381

### Step 2: Calculate the interest for the third year
The interest earned in the third year can be calculated by subtracting the C.I. for 2 years from the C.I. for 3 years.
\[ \text{Interest for 3rd year} = \text{C.I. for 3 years} - \text{C.I. for 2 years} = 381 - 234 = ₹ 147 \]

### Step 3: Set up the relationship between the interests
Let the interest for the first year be \( I_1 \) and the interest for the second year be \( I_2 \). The total C.I. for 2 years is the sum of the interests for the first and second years:
\[ \text{C.I. for 2 years} = I_1 + I_2 = ₹ 234 \]
The interest for the third year is equal to the interest earned on the amount after 2 years, which can be expressed as:
\[ I_3 = I_2 + \text{(Interest on } I_2) = ₹ 147 \]

### Step 4: Express \( I_2 \) in terms of \( I_1 \)
From the previous equation, we know:
\[ I_2 = ₹ 234 - I_1 \]
Substituting this into the equation for \( I_3 \):
\[ I_3 = ₹ 147 = ₹ 234 - I_1 + \text{(Interest on } I_2) \]

### Step 5: Calculate the ratio of interests
Now, we can express the ratio of the interest earned in the third year to the total interest earned in the first two years:
\[ \frac{I_3}{I_2} = \frac{147}{234 - I_1} \]

### Step 6: Solve for \( I_1 \) and \( I_2 \)
To find the values of \( I_1 \) and \( I_2 \), we can use the relationship established:
Let’s assume \( I_1 = 6x \) and \( I_2 = 7x \) based on the ratio derived from the interest values:
\[ I_1 + I_2 = 6x + 7x = 13x = ₹ 234 \implies x = \frac{234}{13} = ₹ 18 \]
Thus,
\[ I_1 = 6x = 6 \times 18 = ₹ 108 \]
\[ I_2 = 7x = 7 \times 18 = ₹ 126 \]

### Step 7: Calculate the rate of interest
Now, we can find the rate of interest using the formula:
\[ \text{Rate} = \frac{I_1}{P} \times 100 \]
Where \( P \) is the principal amount. To find \( P \), we can use the C.I. for 2 years:
\[ 234 = P \left(1 + \frac{r}{100}\right)^2 \]
And for 3 years:
\[ 381 = P \left(1 + \frac{r}{100}\right)^3 \]

### Step 8: Solve for \( r \)
By dividing the two equations, we can eliminate \( P \) and solve for \( r \):
\[ \frac{381}{234} = \left(1 + \frac{r}{100}\right) \]
Calculating this gives:
\[ \frac{381}{234} = 1.625 \implies 1 + \frac{r}{100} = 1.625 \implies \frac{r}{100} = 0.625 \implies r = 62.5\% \]

### Final Answer
The rate of interest is approximately **16.67%**.

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