gmatt1476 wrote:
Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than Democrat members. This year the senate has the same number of members as last year, but it has 2 fewer Republican members than last year. If this year the number of Republican members is 2/3 the number of senate members, how many members does the senate have this year?
(A) 33
(B) 36
(C) 42
(D) 45
(E) 48
PS69400.02
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 membersIn other words: the number of Republicans = (the number of Democrats) + 20
In other words:
R = D + 20This 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 membersImportant: 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 + 6How many members does the senate have this year? We now have the following system:
R = D + 20R = 2D + 6Since both equations are set equal to R, we can write:
D + 20 = 2D + 6Solve to get:
D = 14To find the corresponding value of R, we can take:
R = D + 20Replace D with 14 to get: R = 14 + 20 = 34
So,
R = 34Since 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