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

Question

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

(A) 33 
(B) 36 
(C) 42
(D) 45 
(E) 48

PS69400.02

BrentGMATPrepNow · Accepted Answer

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.

-------------------------------------------------------------------------------------

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

ScottTargetTestPrep · Answer

We can let D = the number of Democrat senators last year; thus, D + 20 = the number of Republican senators last year, D + 2 = the number of Democrat senators this year, and D + 20 - 2 = D + 18 = the number of Republican senators this year. We can create an equation for this year as follows:

(D + 18) / (2D + 20) = 2/3

3(D + 18) = 2(2D + 20)

3D + 54 = 4D + 40

14 = D

Therefore, the total number of senators this year is 2(14) + 20 = 48.

Answer: E

nick1816 · Answer

Number of Republicans last year = R

Number of Democrats last year = D

R-D = 20.....(1)

Number of members this year = R+D

Number of Republicans this year = R-2

Number of Democrats this year = D+2

\frac{R-2}{R+D} = \frac{2}{3}

3R-6 = 2R+2D

R-2D = 6........(2)

Subtract (2) from (1)

D= 14

R= 14+20=34

Total members - R+D =14+34=48

silverprince · Answer

I found that tabulating this problem made my life easier. Here's how:

Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than Democrat members.

Attachment:

1.PNG [ 4.38 KiB | Viewed 48618 times ]

This year the senate has the same number of members as last year,...

Attachment:

2.PNG [ 4.94 KiB | Viewed 48609 times ]

...but it has 2 fewer Republican members than last year.

Attachment:

3.PNG [ 5.66 KiB | Viewed 48539 times ]

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?

So we now know that \(x+18=\frac{2}{3}(2x+20)\) and we need to determine: \(2x+20=?\)

We can now easily solve for x and determine the answer.

chetan2u · Answer

Approach making use of substitution...

Now, last year  R_1=D_1+20 , so total =  D_1+D_1+20=2D_1+20 ....Hence total has to be EVEN......Eliminate A and D

Now take the center value in 36, 42 and 48......
 R_2.. now = 42*\frac{2}{3}=28 , so  R_1=R_2+2=28+2=30  and  D_1=30-20=10 
But  R_1+D_1=30+10=40
eq{42}  Discard

You require the total to be more to increase R and D, so E should be the answer

Now take the center value in 48......
 R_2.. now = 48*\frac{2}{3}=32 , so  R_1=R_2+2=32+2=34  and  D_1=34-20=14 
But  R_1+D_1=34+14=48  ...Our answer

E

Algebraic 
 R_1=D_1+20 , so total =  D_1+D_1+20=2D_1+20 .

R_2=R_1-2=D_1+20-2=D_1+18=\frac{2}{3}*(total)=\frac{2}{3}(2D_1+20) ...

3*(D_1+18)=2(2D_1+20)=4D_1+40........D_1=54-40=14 ..

total =2D_1+20=2*14+20=48

chetan2u · Answer

dave13 
The number of senate members are same in both the years, so does not matter which year you look at.
But be sure that the R and D are for same year, either last or present.

GMATinsight · Answer

Answer: Option E

Video solution by  GMATinsight

https://www.youtube.com/watch?v=YYBFUKDQvBM

IanStewart · Answer

Last year there were 20 more Republicans than Democrats. When 2 Republicans lose seats to Democrats, there will be 16 more Republicans than Democrats. From the 2/3 ratio, we also know there are twice as many Republicans as Democrats. So adding 16 to the number of Democrats is the same as doubling the number of Democrats, and there must therefore be 16 Democrats (because when we double something, we're just adding it to itself), and thus 32 Republicans and 48 Senators.

ahmedmoharam · Answer

I don't know where I am going wrong. Someone please correct me.

Ro is old no. of repub.
Do is old no. of democ.

Rn is new number of repub.
Dn is new number of democ.

S= no. of senate members.

givens:
Ro-20= D0
Ro+Do = S

Rn=Ro-2
Rn= (2/3) S

Rn+Dn=S

Conclusions:

S= Ro+Ro-20=2Ro-20
since Rn=(2/3) S, then Ro-2=(2/3)(2Ro-20) 
Multiply both sides by 3/2
Ro-3= 2Ro-20
then Ro= 36 and Rn=34
34=(2/3) S, then S=34*3/2=51

Why am I getting 51 all of the time?

IanStewart · Answer

Everything looks fine up to the point I've highlighted in red. When you multiply both sides by 3/2, you haven't multiplied the "Ro" on the left by 3/2 (and if you actually solve the highlighted equation as it's currently written, you get Ro = 17 and not Ro = 36). If you solve that equation correctly, you'll get Ro = 34, which is the correct value, and you'll get the right answer from there.

Fdambro294 · Answer

Let the number of Democrats last year = X

Last year the number of Republicans = X + 20

Total Senate last year = 2X + 20

This year’s Total Senate = Last Year’s Total Senate = 2X + 20

This year the Republicans have 2 Fewer = X + 18

And the Republicans make up (2/3) of the this year’s Total Senate

X + 18 = (2/3) (2X + 20)

3X + 54 = 4X + 40

X = 14 = Dems last year

X + 20 = 14 + 20 = 34 Republicans last year

Total senate last year = 14 + 34 = 48

Since the total senate did not change this year, the Total Senate This Year remains 48

(E)
48

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suganyamahendran · Answer

Can some one guide me where exactly am i faltering

last year R - D =20
this year R-D=18 (let us assume that 2 more has gone to D)
x=total members
2/3(x)-1/3(x) =18 --->x=54 total members

Zillions of thanks in advance

BrentGMATPrepNow · Answer

I've highlighted the issue above.

If there are two fewer Republicans this year, then that means there are two additional Democrats. 
So, the number of Republicans this year = R - 2
And, the number of Democrats this year = D + 2
So, this year the number of Republicans is 16 (not 18) greater than the number of Democrats

EMPOWERgmatRichC · Answer

Hi suganyamahendran,

Brent has correctly explained the proper way to 'translate' the wording of the prompt into Algebra. The reason why those translations are correct is because the prompt tells us two details about the state senate:

1) It is composed of ONLY Republicans and Democrats
2) The number of members this year is the SAME as the number of members last year.

Thus, any member who 'switches' to the other group impacts BOTH groups (the original group loses a member (re: -1) and the new group gains a member (re: +1)).

760+: What GMAT Assassins Do To Score at the Highest Levels

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com

MyNameisFritz · Answer

A very nice question!

We are given that Republicans last year are 20 more than Democrats.
The number of democrats = D, Republicans = D+20, total 2D + 20 people in the senate.

This year there are 2 republicans less, therefore D+18
We are said that republicans this year are 2/3 of the total members of the senate (which is still the same member wise (2D+20))

Therefore, (D+18)/(2D+20) = 2/3
Upon solving, 3D+54=4D+40
D=14

Total members in the senate = 2D+20 = 2*14+20=48

Answer E.

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MartyMurray · Answer

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

A simple, intuitive way to answer this one is the following.

First, we can determine how many more Republican members than Democrat members there are this year. We start with what we know about last year.

Last year: R = D + 20

This year, there are 2 fewer Republican members. At the same time, "the senate has the same number of members as last year." So, to for the total to be the same, there must now be 2 more Democrat members.

So, R went down by 2 and D went up by 2, reducing the gap between R and D to 20 - 2 - 2 = 16.

This year: R = D + 16

We know that R is 2/3 of the total. So, D must be 1/3 of the total. Thus, R = 2D.

Thus, D + 16 = 2D.

16 = D

R = 2D. So, the total is the following.

Total: 3D = 3 × 16 = 48

(A) 33
(B) 36
(C) 42
(D) 45
(E) 48

Correct answer:  E

Leon685 · Answer

I'll provide you an alternative approach:

Since we know that this year D = 2/3 of total we also know that R = 1/3 of total.
We also know that this year D-R = 16 --> why? becausce last year the difference was 20 (20 more R than D) and this year we have R-2 and D+2, so difference gets 4 less.
 
D-R = 1/3*total
D-R = 16
16 = 1/3*total | *3
 48 = total

user34745792203 · Answer

Here is an easy approach:

1. Year: R= D + 20

2. Year: R= D + 16
-> 2 fewer Republicans also means 2 more democrats because number of senate members stays the same! 
-> Consequently, the difference between R and D reduces to 16.

Now, the number of Republicans in the senate in the 2. Year is 2/3. 
So, D has to be equal to 1/3. This means 16 has to be 1/3 as well.

Therefore, total senate members: 
=> 16 * 3 = 48

Hope it ́s clear!

TamimHM · Answer

Republic= R
Democrat= D

Last year, 
R= D+20, 
R+D= x

This year, 
D+18+D+2= x
2D= x-20
 D= x-20/2

so, D+18= 2x/3
3x-60+108= 4x
 48=x

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

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