A person borrowed ₹ 1760 at 6% per annum and ₹ 2240 at 9% per annum at simple interest for the same period. He had to pay ₹ 1536 in all as interest. Find the time (in years).

The correct answer is 5.

Given Data:
For the first loan:
Principal ($P_1$) = ₹ 1760
Rate of interest ($R_1$) = 6% per annum

For the second loan:
Principal ($P_2$) = ₹ 2240
Rate of interest ($R_2$) = 9% per annum

Total Simple Interest paid = ₹ 1536
Let the time period for both loans be $T$ years.

Formula:
The formula for Simple Interest ($\text{SI}$) is:

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

Step 1: Calculate the simple interest for the first loan (${\text{SI}}_1$)

${\text{SI}}_1=\frac{1760\times 6\times T}{100}$

${\text{SI}}_1=\frac{10560\times T}{100}=105.6\times T$

Step 2: Calculate the simple interest for the second loan (${\text{SI}}_2$)

${\text{SI}}_2=\frac{2240\times 9\times T}{100}$

${\text{SI}}_2=\frac{20160\times T}{100}=201.6\times T$

Step 3: Set up the equation for Total Simple Interest
The total interest is the sum of interest from both loans:

$\text{Total SI}={\text{SI}}_1+{\text{SI}}_2$

$1536=105.6T+201.6T$

$1536=307.2T$

Step 4: Solve for time ($T$)

$T=\frac{1536}{307.2}$

$T=5$

Thus, the time period for which the money was borrowed is 5 years.