The simple interest on a certain sum is `11(1)/(9)`% of the sum and the number of years is equal to the rate percent per annum. What will be the simple interest on a sum of ₹ 12600 at the same rate for `2 (4)/(5)`years?

Options

A. ₹1216
B. ₹1224
C. ₹1176
D. ₹1164

Answer

To solve the problem step by step, we will break down the information provided and apply the relevant formulas for simple interest.

### Step 1: Understand the Given Information
We know that:
- The simple interest (SI) on a certain sum is \( 11 \frac{1}{9} \% \) of the sum.
- The number of years (T) is equal to the rate percent (R) per annum.

### Step 2: Convert the Percentage to a Fraction
First, convert \( 11 \frac{1}{9} \% \) into a fraction:
\[ 11 \frac{1}{9} = \frac{100}{9} \% \]
This means that the simple interest is \( \frac{100}{9} \% \) of the principal (P).

### Step 3: Set Up the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Since \( R = T \), we can rewrite this as:
\[ SI = \frac{P \times T \times T}{100} = \frac{P \times T^2}{100} \]

### Step 4: Relate Simple Interest to the Given Percentage
From the information given, we know that:
\[ SI = \frac{100}{9} \% \text{ of } P = \frac{100}{9} \times \frac{P}{100} = \frac{P}{9} \]
Now we have two expressions for SI:
1. \( SI = \frac{P \times T^2}{100} \)
2. \( SI = \frac{P}{9} \)

### Step 5: Set the Two Expressions Equal
Equating the two expressions for SI:
\[ \frac{P \times T^2}{100} = \frac{P}{9} \]
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{T^2}{100} = \frac{1}{9} \]

### Step 6: Solve for T
Cross-multiplying gives:
\[ 9T^2 = 100 \]
\[ T^2 = \frac{100}{9} \]
Taking the square root:
\[ T = \frac{10}{3} \text{ years} \]

### Step 7: Calculate Simple Interest for the Given Principal
Now, we need to calculate the simple interest on a sum of ₹ 12,600 for \( 2 \frac{4}{5} \) years at the same rate. First, convert \( 2 \frac{4}{5} \) years into an improper fraction:
\[ 2 \frac{4}{5} = \frac{14}{5} \text{ years} \]

### Step 8: Determine the Rate
Since \( T = \frac{10}{3} \) years, the rate \( R \) is also \( \frac{10}{3} \% \).

### Step 9: Apply the Simple Interest Formula
Using the formula for simple interest:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting \( P = 12600 \), \( R = \frac{10}{3} \), and \( T = \frac{14}{5} \):
\[ SI = \frac{12600 \times \frac{10}{3} \times \frac{14}{5}}{100} \]

### Step 10: Simplify the Calculation
Calculating the expression:
\[ SI = \frac{12600 \times 10 \times 14}{3 \times 5 \times 100} \]
\[ = \frac{12600 \times 140}{1500} \]
\[ = \frac{1764000}{1500} = 1176 \]

### Final Answer
The simple interest on the sum of ₹ 12,600 at the same rate for \( 2 \frac{4}{5} \) years is ₹ 1176.