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 the schemes is ₹3717. If both schemes have a 13% per annum interest rate, then what is the first-year interest (in ₹) for the simple interest scheme?

Let the principal sum invested in each scheme be P.
Rate of interest (R) =13% per annum.
Time (T) =2 years.

Simple Interest (SI) for 2 years:

SI=
100
P×R×T
​
=
100
P×13×2
​
=
100
26P
​

Compound Interest (CI) for 2 years:

CI=P[(1+
100
R
​
)
2
−1]=P[(1+
100
13
​
)
2
−1]
CI=P[(
100
113
​
)
2
−1]=P[
10000
12769
​
−1]=
10000
2769P
​

Given that the total interest is ₹3717:

SI+CI=3717
100
26P
​
+
10000
2769P
​
=3717
10000
2600P+2769P
​
=3717
10000
5369P
​
=3717
P=
5369
3717×10000
​

Dividing 3717 by 5369:

P=0.6925×10000⟹P=6925 (approx evaluation)
Let's re-verify the multiplier: 5369×0.7=3758.3. Let's check
5369
3717
​
:
Actually, 3717/5369≈0.6923. Let's re-calculate using effective percentage rates:

Effective SI rate for 2 years =13%+13%=26%

Effective CI rate for 2 years =13+13+
100
13×13
​
=26+1.69=27.69%

Total combined interest rate =26%+27.69%=53.69%

So,

53.69% of P=3717
100
53.69
​
×P=3717⟹P=
53.69
3717×100
​
=6924.9...
Wait, let's look at the options. First year interest for simple interest is 13% of P.
If first year interest is I, then:

Total SI =2I

Total CI =2I+I×13%=2I+0.13I=2.13I

Total interest =2I+2.13I=4.13I

Given:

4.13I=3717
I=
4.13
3717
​
=900
Thus, the first-year interest for the simple interest scheme is ₹900.