Last year a state senate consisting of only Republican and Democrat me

STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test the answer choices.
Now let's give ourselves up to 20 seconds (max) to identify a faster approach.
In this case, we can also solve the question algebraically, but testing the answer choices is easier.

Let's start with answer choice C:
(C) 42. So there are 42 Senate members.
2/3 are Republicans, which means there are 28 Republicans and 14 Democrats THIS YEAR
So, LAST YEAR there were 30 Republicans and 12 Democrats, which means there were 18 more Republican members than Democrat members.
The question tells us that, last year, there were 20 more Republican members than Democrat members, which means answer choice C is incorrect.

Since we want to increase the difference from 18 more Republicans to 20 more Republicans, we need more than 42 senate members, which means we can also eliminate answer choices A and B.

At this point we need only test one more answer choice. If we test choice D, and it's correct, then the correct answer is D. If choice D is incorrect, then the correct answer must be E.
Let's test choice D
(D) 45. So there are 45 Senate members.
2/3 are Republicans, which means there are 30 Republicans and 15 Democrats THIS YEAR
So, LAST YEAR there were 32 Republicans and 13 Democrats, which means there were 19 more Republican members than Democrat members.
The question tells us that, last year, there were 20 more Republican members than Democrat members, which means answer choice D is incorrect.

By the process of elimination, the correct answer is E.

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ALGEBRAIC APPROACH:
Let D = the number of Democrats LAST YEAR
Let R = the number of Republicans LAST YEAR

Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than Democrat members
In other words: the number of Republicans = (the number of Democrats) + 20
In other words: R = D + 20

This year the senate has the same number of members as last year, but it has 2 fewer Republican members than last year.
So, R - 2 = the number of Republicans THIS YEAR

Important: Since the Senate still has the same number of members as last year, there must be 2 additional Democrats this year (to account for the fact there are 2 fewer Republicans).
So, D + 2 = the number of Democrats THIS YEAR

If this year the number of Republican members is 2/3 the number of senate members
Important: If Republicans account for 2/3 of the senate, then Democrats must account for the remaining 1/3 of senate. Since 2/3 is TWICE 1/3, we can say: (the number of Republicans this year) = 2(the number of Democrats this year).
In other words: R - 2 = 2(D + 2)
Expand the right side: R - 2 = 2D + 4
Add 2 to both sides to get: R = 2D + 6

How many members does the senate have this year?
We now have the following system:
R = D + 20
R = 2D + 6

Since both equations are set equal to R, we can write: D + 20 = 2D + 6
Solve to get: D = 14

To find the corresponding value of R, we can take: R = D + 20
Replace D with 14 to get: R = 14 + 20 = 34
So, R = 34

Since R + D = the total number of senators LAST YEAR, and since the Senate has the same number of members THIS YEAR, we can conclude that:
The total number of Senators THIS YEAR = R + D = 14 + 34 = 48

Answer: E

Cheers,
Brent