The simple interest on a certain sum at 14% per annum for ` 3 (1)/(4)` years is ₹3731. What will the amount of the same sum for `5(1)/(2)` years at half the earlier rate?

Options

A. ₹11931
B. ₹11913
C. ₹11537
D. ₹11357

Answer

To solve the problem step by step, we will follow the process of calculating the principal amount first and then the amount for the new time and rate.

### Step 1: Identify the given information
- Simple Interest (SI) = ₹3731
- Rate (R) = 14% per annum
- Time (T) = 3 1/4 years = 13/4 years

### Step 2: Use the Simple Interest formula
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 3731 = \frac{P \times 14 \times \frac{13}{4}}{100} \]

### Step 4: Simplify the equation to find the principal (P)
Rearranging the equation gives:
\[ P = \frac{3731 \times 100}{14 \times \frac{13}{4}} \]

Calculating the denominator:
\[ 14 \times \frac{13}{4} = \frac{182}{4} = 45.5 \]

Now substituting back:
\[ P = \frac{3731 \times 100}{45.5} \]

Calculating \( P \):
\[ P = \frac{373100}{45.5} = 8200 \]

### Step 5: Calculate the new Simple Interest for 5 1/2 years at half the earlier rate
- New Time (T2) = 5 1/2 years = 11/2 years
- New Rate (R2) = 14% / 2 = 7%

Using the Simple Interest formula again:
\[ SI2 = \frac{P \times R2 \times T2}{100} \]

Substituting the values:
\[ SI2 = \frac{8200 \times 7 \times \frac{11}{2}}{100} \]

### Step 6: Simplify to find SI2
Calculating:
\[ SI2 = \frac{8200 \times 7 \times 11}{200} \]
\[ = \frac{634600}{200} = 3173 \]

### Step 7: Calculate the total amount
The total amount (A) is given by:
\[ A = P + SI2 \]
Substituting the values:
\[ A = 8200 + 3173 = 11373 \]

### Final Answer
The amount of the same sum for 5 1/2 years at half the earlier rate will be ₹11373.

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