A sum becomes 10/9 times itself in 1 yr Find the rate of simple interest

Options

A. `11(1)/(2) %`
B. `11(1)/(9) %`
C. `12 (1)/(2) %`
D. `12(1)/(2) %`

Answer

To find the rate of simple interest when a sum becomes \( \frac{10}{9} \) times itself in 1 year, we can follow these steps:

### Step 1: Define the Principal Amount
Let the principal amount be \( P \).

### Step 2: Determine the Amount After 1 Year
According to the problem, after 1 year, the amount becomes \( \frac{10}{9} P \).

### Step 3: Calculate the Simple Interest
The formula for simple interest (SI) is:
\[ SI = \text{Amount} - \text{Principal} \]
Substituting the values we have:
\[ SI = \frac{10}{9} P - P \]
To simplify this, we can express \( P \) as \( \frac{9}{9} P \):
\[ SI = \frac{10}{9} P - \frac{9}{9} P = \frac{1}{9} P \]

### Step 4: Use the Simple Interest Formula
The formula for simple interest is also given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years. Here, \( T = 1 \) year, so we can rewrite the equation as:
\[ SI = \frac{P \times R \times 1}{100} = \frac{P \times R}{100} \]

### Step 5: Set the Two Expressions for SI Equal
Now we can set the two expressions for simple interest equal to each other:
\[ \frac{1}{9} P = \frac{P \times R}{100} \]

### Step 6: Cancel \( P \) from Both Sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{1}{9} = \frac{R}{100} \]

### Step 7: Solve for \( R \)
To find \( R \), we can cross-multiply:
\[ 1 \times 100 = 9 \times R \]
This simplifies to:
\[ 100 = 9R \]
Now, divide both sides by 9:
\[ R = \frac{100}{9} \]

### Step 8: Final Result
Thus, the rate of simple interest is:
\[ R \approx 11.11\% \]