Problem Solving

BTGModeratorVI wrote: ↑

Thu Nov 05, 2020 7:56 am

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

Answer: E
Source: Official guide

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

BTGModeratorVI wrote: ↑

Thu Nov 05, 2020 7:56 am

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

Answer: 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.

Answer: E