A finance company advances Rs. 12,000 at 7% per annum simple interest. It also provides Rs. 8,000 at 11% per annum simple interest for the same duration. If the combined simple interest earned is Rs. 6,880, determine the duration of both loans.

The correct answer is 4 years.

Step 1: Identify the given data

Let the duration of both loans be T years.

For the first loan:
Principal (P₁) = Rs. 12,000
Rate of interest (R₁) = 7% per annum

For the second loan:
Principal (P₂) = Rs. 8,000
Rate of interest (R₂) = 11% per annum

Total simple interest earned (SI_total) = Rs. 6,880

The standard formula for simple interest (SI) is:

$SI=\frac{P\times R\times T}{100}$

Step 2: Calculate the simple interest earned from each loan

Simple interest from the first loan (SI₁):

$S{I}_1=\frac{12000\times 7\times T}{100}=840T$

Simple interest from the second loan (SI₂):

$S{I}_2=\frac{8000\times 11\times T}{100}=880T$

Step 3: Formulate and solve the equation for total interest

Summing the interest from both loans gives the combined simple interest:

$S{I}_{total}=S{I}_1+S{I}_2$

Substitute the expressions into the total interest equation:

$6880=840T+880T$

$6880=1720T$

Now, divide both sides by 1,720 to find T:

$T=\frac{6880}{1720}$

$T=4$

Therefore, the duration of both loans is 4 years.