The simple interest on a certain sum in `5(1)/(2)` years at 6% p.a. is ₹ 2541. what will be the amount of the same sum in `5(2)/(3)` years at `8(1)/(7)`% p.a.?

Options

A. ₹13125
B. ₹11235
C. ₹12153
D. ₹11253

Answer

To solve the problem step by step, we will use the formula for simple interest and the relationship between principal, interest, and amount.

### Step 1: Understand the Simple Interest Formula
The formula for calculating simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Convert Time to a Fraction
We are given the time as \( 5 \frac{1}{2} \) years. We convert this to an improper fraction:
\[ 5 \frac{1}{2} = \frac{11}{2} \text{ years} \]

### Step 3: Substitute Known Values to Find Principal
We know:
- \( SI = 2541 \)
- \( R = 6\% \)
- \( T = \frac{11}{2} \)

Substituting these values into the formula:
\[ 2541 = \frac{P \times 6 \times \frac{11}{2}}{100} \]

### Step 4: Simplify the Equation
Rearranging the equation to solve for \( P \):
\[ 2541 = \frac{P \times 6 \times 11}{200} \]
Multiplying both sides by 200:
\[ 2541 \times 200 = P \times 66 \]
\[ 508200 = P \times 66 \]
Now, divide both sides by 66:
\[ P = \frac{508200}{66} = 7700 \]

### Step 5: Calculate the Amount for New Time and Rate
Now we need to find the amount after \( 5 \frac{2}{3} \) years at \( 8 \frac{1}{7}\% \) per annum. First, convert these values:
- Time: \( 5 \frac{2}{3} = \frac{17}{3} \) years
- Rate: \( 8 \frac{1}{7} = \frac{57}{7}\% \)

### Step 6: Calculate Simple Interest for New Time and Rate
Using the principal \( P = 7700 \):
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the new values:
\[ SI = \frac{7700 \times \frac{57}{7} \times \frac{17}{3}}{100} \]

### Step 7: Simplify the Calculation
Calculating the expression:
\[ SI = \frac{7700 \times 57 \times 17}{7 \times 3 \times 100} \]
Calculating the denominator:
\[ 7 \times 3 \times 100 = 2100 \]
Calculating the numerator:
\[ 7700 \times 57 \times 17 = 7700 \times 969 = 7461300 \]
Now, divide:
\[ SI = \frac{7461300}{2100} = 3553 \]

### Step 8: Calculate the Total Amount
The total amount \( A \) is given by:
\[ A = P + SI \]
Substituting the values:
\[ A = 7700 + 3553 = 11253 \]

### Final Answer
The amount after \( 5 \frac{2}{3} \) years at \( 8 \frac{1}{7}\% \) per annum is ₹ 11253.

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