It is of course possible to modelise the whole thing but I think that number picking is much more efficient here.

So let's assume Henry has 50 he gets 2 dollars for each dollar saved this year, then the solution is that he should spend 40 this year, save 10. Next year, he will have 20 which is 50% of 40. So the answer that we want is 1/5

Let's evaluate the potential answers:
A) 1 / R+2 = 1/3 --> OUT
B) 1 / 2R+2 = 1/4 --> OUT
C) 1/ 3R+2 = 1/5 --> OK
D) 1 / R+3 = 1/4 --> OUT
E) 1 / 2R+3 = 1/5 --> OK

Let's try another example:
If Henry has 80 and he gets 1.5 dollar for each dollar saved this year, then the solution is that he should save 60 so he keeps 20. Next year, he will have 30, which is 50% of 60. So the answer we want is 1/4.

C) 1 / 3R+2 = 1 / (3/2 + 2) = 1 / 7/2 = 2/7 --> OUT
E) 1 / 2R + 3 = 1/ (2/2 + 3) = 1 / (1+3) = 1/4 --> OK

So I think ans is E

And here is how to do it in a 'modelised' way.

If Henry has a 100, he will spend X and save Y.

We know that he will need to have half as much next year as this year. Therefore:
X = 2 Y (1+R)

We are looking for the share of the total that he needs to save:

Y / (X+Y) = Y / (2Y(1+R) + Y) = 1 / (2 + 2R + 1) = 1/(2R+3) --> ANS E