Two equal sums are invested in two different schemes. One scheme gives simple interest and the other gives compound interest (annual compounding). The sum of interest obtained after 2 years from both schemes is Rs 3717. If both schemes have a 13% per annum interest rate, then what is the first-year interest (in Rs) for the simple interest scheme?

Let the sum invested in each scheme be P.
The rate of interest is r=13% per annum.

1. Simple Interest (SI) after 2 years:

SI=
100
P×r×t
​
=
100
P×13×2
​
=
100
26P
​
=0.26P
The interest for the first year under SI is half of the total SI:

SI
1st year
​
=
100
13P
​
=0.13P
2. Compound Interest (CI) after 2 years:
The effective CI rate for 2 years at 13% per annum is given by successive percentage addition:

Effective rate=13+13+
100
13×13
​
=26+1.69=27.69%
CI=27.69% of P=
100
27.69P
​
=0.2769P
3. Sum of interests:
Given that the sum of interest from both schemes is Rs 3717:

SI+CI=3717
0.26P+0.2769P=3717
0.5369P=3717
P=
0.5369
3717
​
=7000
So, the principal sum is Rs 7000.

4. First year interest for simple interest scheme:

SI
1st year
​
=13% of 7000=
100
13
​
×7000=900