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

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

by BTGmoderatorDC » Tue Jun 30, 2020 7:25 pm

00:00

A

B

C

D

E

## Global Stats

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

OA E

Source: Official Guide

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

by [email protected] » Wed Jul 01, 2020 12:31 am
BTGmoderatorDC wrote:
Tue Jun 30, 2020 7:25 pm
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

OA E

Source: Official Guide
Say, last year, there were R no. of republicans and D no. of Democrats, so the total no. of members = R + D.

Also, we know that R = D + 2 ---(1)

Given that this year, the no. of members is the same as that in the last year, so, this year, the no.of members = (R + D)

Also, we know that No. of republicans = R – 2, so the no. of Democrats = D + 2.

Also, it is given that R – 2 = 2/3 of (R + D)

From equation (1), we get

R – 2 = 2/3 of (R + D)
D + 20 – 2 = 2/3 of (D + 20 + D)

D = 14 and R = 34, so R + D = 14 + 34 = 48

Hope this helps!

-Jay
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### Re: Last year a state senate consisting of only Republican and Democrat members had 20 more Republican members than

by [email protected] » Wed Apr 21, 2021 6:37 am
BTGmoderatorDC wrote:
Tue Jun 30, 2020 7:25 pm
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

OA E

Source: Official Guide
Solution:

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.

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

by [email protected] » Fri Apr 08, 2022 6:13 am
BTGmoderatorDC wrote:
Tue Jun 30, 2020 7:25 pm
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

OA E

Source: Official Guide
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.

(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