Ayush invests some amount of money at the rate of 10% p.a. After 2 years he invests 2 times the previous amount of money at an interest rate of 15% p.a. At the end of 5 years, the total interest earned is Rs. 19,600. For how many years should Ayush invest the same amount of money at the rate of 5% p.a. to earn the interest of Rs. 39,200?

Let's break down the problem step by step:

Step 1: Understand the problem Ayush invests an amount (let's call it P) at 10% per annum for 2 years. Then he invests 2P at 15% per annum for the remaining 3 years (since the total time is 5 years). The total interest earned from these investments is Rs. 19,600. We need to find out how long he should invest the same amount (P) at 5% per annum to earn Rs. 39,200.

Step 2: Calculate the interest earned from the first investment The formula for simple interest is I = PRT/100, where I is the interest, P is the principal amount, R is the rate of interest, and T is the time in years. So, the interest from the first investment is I1 = P102/100 = P/5.

Step 3: Calculate the interest earned from the second investment The interest from the second investment is I2 = 2P153/100 = 9P/10.

Step 4: Calculate the total interest The total interest earned is I1 + I2 = P/5 + 9P/10 = Rs. 19,600.

Step 5: Solve for P Solving the equation from step 4 gives P = Rs. 8,000.

Step 6: Calculate the time needed to earn Rs. 39,200 at 5% per annum Using the simple interest formula again, we have T = I/PR100 = 39,200/(8000*5)*100 = 9.8 years.

So, Ayush should invest Rs. 8,000 for approximately 9.8 years at 5% per annum to earn Rs. 39,200.