A certain sum amounts to ₹11760 in `2(1)/(2)` years at 9% p.a. simple interest. What will be the simple interest on the same sum for `4(2)/(5)` years at 15% p.a?

Options

A. ₹6336
B. ₹6363
C. ₹6436
D. ₹6433

Answer

To solve the problem step by step, we need to find the principal amount first and then calculate the simple interest for the new time period and rate.

### Step 1: Understand the given information
We know that the amount (A) after a certain time (T) at a certain rate (R) can be calculated using the formula:
\[ A = P + SI \]
Where:
- \( A \) = Final amount (₹11760)
- \( P \) = Principal amount
- \( SI \) = Simple Interest

### Step 2: Calculate the Simple Interest for the first scenario
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]

Given:
- \( R = 9\% \)
- \( T = 2.5 \) years (which is \( \frac{5}{2} \))

We can rewrite the total amount as:
\[ A = P + SI \]
So,
\[ 11760 = P + \frac{P \times 9 \times 2.5}{100} \]

### Step 3: Substitute and simplify
Substituting the values into the equation gives:
\[ 11760 = P + \frac{P \times 22.5}{100} \]
\[ 11760 = P + \frac{22.5P}{100} \]
\[ 11760 = P \left(1 + \frac{22.5}{100}\right) \]
\[ 11760 = P \left(\frac{100 + 22.5}{100}\right) \]
\[ 11760 = P \left(\frac{122.5}{100}\right) \]

### Step 4: Solve for the Principal (P)
Now, we can solve for \( P \):
\[ P = \frac{11760 \times 100}{122.5} \]
\[ P = \frac{1176000}{122.5} \]
\[ P = 9600 \]

### Step 5: Calculate the Simple Interest for the new scenario
Now we need to find the simple interest for the same principal amount at a new rate and time:
- New \( R = 15\% \)
- New \( T = 4.5 \) years (which is \( \frac{22}{5} \))

Using the SI formula again:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{9600 \times 15 \times 4.5}{100} \]

### Step 6: Calculate the Simple Interest
\[ SI = \frac{9600 \times 15 \times 4.5}{100} \]
\[ SI = \frac{9600 \times 67.5}{100} \]
\[ SI = \frac{645000}{100} \]
\[ SI = 6450 \]

### Final Answer
The simple interest on the same sum for \( 4\frac{2}{5} \) years at 15% p.a. is ₹6450.

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