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

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
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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 \(R_1+D_1=30+10=40\neq{42}\) 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 = \(D_1+D_1+20=2D_1+20\).

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

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

\(total =2D_1+20=2*14+20=48 \)
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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

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.
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thanks chetan2u but one logical thing i don`t get is this

even if number of members remained the same yet the ratio between D to R changed this year i.e. the number of republicans decreased by 2 this year, .... so if it decreased by 2 then one can not find number of republicans this year by equating data from this year to that of last year .... though yeah you still get the answer

2/3 is # of R this year

and D+20 is number of R last year (D+D+20) total number of members last year

this year number of R decreased by 2 so how one can write 2/3*(2D+20) ... even if 2d+20 is the same number of members it has different number/ratio of R to D

i think there must be a typo in a wording of this question, no not in mine but in an official og

dave13
There's no problem. Here \(R_L + D_L = R_T' + D_T' = 48\)
Now \(R_L = D_L + 20 = R_T' + 2\)
on solving \(D_L = 14, R_L = 34, R_T' = 32\) and \(D_T' = 16\)

It is valid since \(32 = \frac{2}{3}*(32+16)\) which as per question is right \(R_T' = \frac{2}{3}* (R_T' + D_T')\)
_________________

What I do not get is why this year R-2 = D+2 ? I made a mistake writing it as R-2 = D+20. Can you please elaborate more ?


thanks,
Vesper

in the question it is nowhere mentioned that R-2 = D+2; they have mentined that last year if Republicans = R, then this year it will be R-2. So Democrats will be D+2 (D: Democrats last year) but R-2 is not equal to D+2.
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I'm not sure what you're referring to.
I don't see anywhere in my solution where I say R-2 = D+2
Can you please elaborate?

THANK YOU! This saved me from a headache.

I have no idea why I didn't think of it in terms of a ratio.

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.


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?

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.

Step 1: Understand (Glance Read Jot)

This Algebraic Translation problem asks you to solve for the number of members in the senate. Information is given about Republicans and Democrats in the senate this year and last year. Working Backwards may be possible because the answer choices represent a simple value (i.e., the total members) and are integers, making them relatively easy to work with.

Jot down the given information.

Last year: 20 more R’s than D’s

This year: 2 fewer R’s than last year, R’s = 23 Total

Total?



Step 2: Plan (Reflect Organize)

As you consider whether to work backwards, think about whether the answer choices (the total number of members) would be easy to plug into the given information. Since the total number of members is the same both years, working backwards will be a good strategy.

Alternatively, you could translate the given information into equations and solve.



Step 3: Solve (Do the Work)

Work Backwards

Start with answer choice (B). Plug in 36 for the total number of members, and solve for the Republicans each year.

This Year

If there are 36 total members, 36(23) = 24 Republicans this year.

Last Year

Last year, there were 2 more Republicans, meaning there were 26 Republicans.

If there were 26 Republicans last year, there were 36 −26 = 10 Democrats.

The difference between the Republicans and Democrats is 26 − 10 = 16. But according to the problem, this difference should be 20. Since the difference is too small, test a larger answer.

Test answer choice (D).

This Year

If there are 45 total members, 45(23) = 30 Republicans this year.

Last Year

Last year, there were 2 more Republicans, meaning there were 32 Republicans.

If there were 32 Republicans last year, there were 45 −32 = 13 Democrats.

The difference between the Republicans and Democrats is 32 − 13 = 19. But according to the problem, this difference should be 20. Since the difference is still too small, select the only larger answer.

Algebra

For the algebraic solution, please see the Official Guide.

The correct answer is (E).

Hit Kudos, if it is useful.

Answer: Option E

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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.
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let Republicans last year be x and Democrat = y
given
x=y+20
so x+y ; y+20+y ; 2y+20
this year
R = x-2 ; y+20-2 ; y-18
(y-18) = 2/3 * ( 2y+20)
solve for y
y= 14
we get x = 34
x+y ; 14+34 ; 48
option E

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?

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.
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Thanks for the reply.

I found my mistake, but its not that I didn't multiply Ro by (3/2). what I did was:
(3/2) Ro-3= 2Ro-20
Then 2Ro- (3/2) = 18 not 17.

Your clarification pointed out my silly mistake.