The simple interest on a certain sum for 4 years at 14% per annum is ₹6,160 less than the simple interest on the same sum at the same rate for 8 years. Find the sum.

The correct option is ₹11,000.

Let's break down the solution step-by-step.

Step 1: Understand the formula for Simple Interest (SI)
The formula to calculate simple interest is given by:
$\mathrm{SI}=\frac{P\times R\times T}{100}$
where:

• $P$ is the principal amount (the sum of money)
• $R$ is the rate of interest per annum
• $T$ is the time period in years

Step 2: Set up the equations based on the given information
Let the sum (principal) be $P$ rupees.
The rate of interest ($R$) is given as 14% per annum for both cases.

Case 1: Simple interest on the sum for 8 years ($T_1=8$ years):
${\mathrm{SI}}_1=\frac{P\times 14\times 8}{100}$

Case 2: Simple interest on the same sum for 4 years ($T_2=4$ years):
${\mathrm{SI}}_2=\frac{P\times 14\times 4}{100}$

Step 3: Use the relationship given in the question
According to the problem, the simple interest for 4 years is ₹6,160 less than the simple interest for 8 years:
${\mathrm{SI}}_1-{\mathrm{SI}}_2=6160$

Substitute the values of ${\mathrm{SI}}_1$ and ${\mathrm{SI}}_2$ into the equation:
$\frac{P\times 14\times 8}{100}-\frac{P\times 14\times 4}{100}=6160$

Step 4: Solve for $P$
Factor out the common terms on the left-hand side:
$\frac{P\times 14}{100}\times (8-4)=6160$

Simplify the term in the parenthesis:
$\frac{P\times 14\times 4}{100}=6160$

Multiply the numbers in the numerator:
$\frac{56P}{100}=6160$

Isolate $P$:
$P=\frac{6160\times 100}{56}$

Calculate the value:
$P=110\times 100$
$P=11000$

Thus, the sum is ₹11,000.