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

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Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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
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Last year a state senate consisting of only Republican and Democrat me [#permalink]
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gmatt1476
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

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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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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gmatt1476
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

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 = $$D_1+D_1+20=2D_1+20$$.

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

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

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

chetan2u why did you write $$D_1+18=\frac{2}{3}*(total)=\frac{2}{3}(2D_1+20)$$ and NOT

$$D+18=\frac{2}{3}*(total)=\frac{2}{3}(D + 18)$$ if question asks # of members this year

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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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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 36170 times ]

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

2.PNG [ 4.94 KiB | Viewed 36209 times ]

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

3.PNG [ 5.66 KiB | Viewed 36191 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.
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Last year a state senate consisting of only Republican and Democrat me [#permalink]
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gmatt1476
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

Video solution by GMATinsight

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Q51  V47
Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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?
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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
ahmedmoharam
I don't know where I am going wrong. Someone please correct me.

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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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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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Last year a state senate consisting of only Republican and Democrat me [#permalink]
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

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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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suganyamahendran
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

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
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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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)).

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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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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

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Re: Last year a state senate consisting of only Republican and Democrat me [#permalink]
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­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 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