A sum of money put out at compound interest amounts in 2 years to Rs. 578·40 and in 3 years to Rs. 614·55. Find the rate of interest.

Options

A. `6%`
B. `(6)1/4 %`
C. `(6)1/2 %`
D. `(6)3/4%`

Answer

To find the rate of interest, we can use the information given about the amounts after 2 years and 3 years.

Let’s denote:
- Amount after 2 years (A2) = Rs. 578.40
- Amount after 3 years (A3) = Rs. 614.55

### Step 1: Calculate the interest earned in the third year
The interest earned in the third year can be calculated by subtracting the amount after 2 years from the amount after 3 years.

\[ \text{Interest for 3rd year} = A3 - A2 = 614.55 - 578.40 = 36.15 \]

### Step 2: Calculate the principal amount
The amount after 2 years (A2) can be expressed in terms of the principal (P) and the rate of interest (r):

\[ A2 = P(1 + r)^2 \]

The amount after 3 years (A3) can be expressed as:

\[ A3 = P(1 + r)^3 \]

### Step 3: Set up the equation for the third year
From the interest earned in the third year, we can express it as:

\[ \text{Interest for 3rd year} = A3 - A2 = P(1 + r)^3 - P(1 + r)^2 \]

Factoring out \(P(1 + r)^2\):

\[ 36.15 = P(1 + r)^2 \cdot r \]

### Step 4: Substitute the value of A2 into the equation
From the expression for A2, we can express \(P(1 + r)^2\):

\[ P(1 + r)^2 = 578.40 \]

### Step 5: Substitute into the interest equation
Now we can substitute this into the interest equation:

\[ 36.15 = 578.40 \cdot r \]

### Step 6: Solve for r
Now, we can solve for r:

\[ r = \frac{36.15}{578.40} \]

Calculating this gives:

\[ r \approx 0.0625 \]

### Step 7: Convert r to percentage
To convert the rate into percentage, we multiply by 100:

\[ \text{Rate of interest} = r \times 100 = 0.0625 \times 100 = 6.25\% \]

### Final Answer:
The rate of interest is **6.25%**.

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