Solution:

Given,

Amount after 3 years = Rs. 500

Amount after 5 years = Rs. 600

Let the principal be P and the rate of interest be R.

Then,

Simple Interest for 3 years = (P * R * 3)/100

Simple Interest for 5 years = (P * R * 5)/100

We can write two equations using the given information:

P + (P * R * 3)/100 = 500

P + (P * R * 5)/100 = 600

Solving these equations, we get:

P = Rs. 400

Hence, the principal is Rs. 400.

Explanation:

To solve this problem, we need to use the formula for simple interest, which is:

Simple Interest = (P * R * T)/100

Where P is the principal, R is the rate of interest, and T is the time period.

We have two equations and two variables (P and R), so we can solve for P.

We can simplify the equations by multiplying both sides by 100:

100P + 3PR = 50000

100P + 5PR = 60000

We can then eliminate P by subtracting the first equation from the second equation:

2PR = 10000

PR = 5000

We can substitute this value of PR into either of the original equations to solve for P.

P + (5000 * 3)/100 = 500

P + 150 = 500

P = 350

Or,

P + (5000 * 5)/100 = 600

P + 250 = 600

P = 350

However, we made a mistake by assuming that the interest rate is the same for both time periods. Since the interest is simple interest, the interest rate can vary over time.

Instead of assuming a fixed interest rate, we can use the given information to set up two equations in terms of P and solve for P.

P + (P * R * 3)/100 = 500

P + (P * R * 5)/100 = 600

We can subtract the first equation from the second equation to eliminate P:

(P * R * 2)/100 = 100

PR = 5000

Substituting this value of PR into the first equation, we get:

P + 150 = 500

P = 350

This answer is incorrect because we used the wrong interest rate.

To correct our mistake, we can use a different approach.

We know that the interest earned in the first three years is Rs. 100 (since the amount increased from Rs. 400 to Rs. 500).

Therefore, the interest earned in the next two years is Rs. 100 (since the amount increased from Rs. 500 to Rs. 600).

The total interest earned over five years is Rs. 200.

Using the formula for simple interest, we can set up an equation in terms of P and solve for P:

(P * R * 5)/100 = 200

PR = 4000

Substituting this value of PR into either of the original equations, we get:

P + 60 = 200

P = 140

Or,

AMT @ the end of 5 yrs = 600.
AMT @ the end of 3 yrs = 500.
Int. for the 2yrs (5-3 yrs) =100.
Int for 1yr =100/2 (frm above) =50 .
as we r GN with the details fr 3 yrs we find the int AMT fr 3 yrs.,ie., int fr 3yrs= 50*3=150 .
therefore P=AMT -Int = 500-150= 350 .
u can also find r% by r= 100 * int ÷ principal * n , here it is 100 * 150÷ 350*3 = 14.2857%.
Now u can also verify this by using Int = pnr÷100 .
int = 350*3*14.2857÷100= 149.99985 =~ 150 ( int fr 3 yrs as GN in question) .
therefore the answer found above is crt.