If some amount becomes `(25)/(16)`of itself in 2 years at certain rate of compound interest then find the rate of interest?

Options

A. `24%`
B. `26%`
C. `27%`
D. `25%`

Answer

To solve the problem step by step, we will use the formula for compound interest and the information given in the question.

### Step 1: Understand the Problem
We are given that an amount becomes \( \frac{25}{16} \) of itself in 2 years at a certain rate of compound interest. We need to find the rate of interest.

### Step 2: Set Up the Equation
Let the principal amount be \( P \). According to the problem, the amount after 2 years is:
\[ A = \frac{25}{16} P \]
The formula for the amount \( A \) in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where \( R \) is the rate of interest and \( t \) is the time in years.

### Step 3: Substitute Values
Substituting the values into the formula, we have:
\[ \frac{25}{16} P = P \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{25}{16} = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Take the Square Root
To eliminate the square, we take the square root of both sides:
\[ \sqrt{\frac{25}{16}} = 1 + \frac{R}{100} \]
Calculating the square root gives:
\[ \frac{5}{4} = 1 + \frac{R}{100} \]

### Step 6: Solve for \( R \)
Now, we can isolate \( R \):
\[ \frac{5}{4} - 1 = \frac{R}{100} \]
Calculating \( \frac{5}{4} - 1 \):
\[ \frac{5}{4} - \frac{4}{4} = \frac{1}{4} \]
So, we have:
\[ \frac{1}{4} = \frac{R}{100} \]

### Step 7: Multiply to Find \( R \)
Now, we multiply both sides by 100 to find \( R \):
\[ R = 100 \times \frac{1}{4} = 25 \]

### Conclusion
Thus, the rate of interest \( R \) is:
\[ \boxed{25\%} \]

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