William H. answered • 04/21/20

Patient and Effective SAT Math and English Tutor

Since the presence of Independents is not a condition for forming a committee, we can ignore the information given about Independents.

Since an infinite number of committees can be formed if members are not limited in the number of committees on which they can serve, and we are given no information about such limitations, let us assume that members can serve on only one committee.

Since there are 54 Republicans, and three Republicans must serve on a given committee, the number of committees, x, that can be formed will equal the number of groups of three in 54. So, x = 54/3 = 18. If there had been a remainder, we would ignore the remainder because it would be too few Republicans to form an additional committee.

Writing the equation out in full, we have: x = 54 Republicans/(3 Republicans per committee) = 54 Republicans/(3 Republicans/committee) = 18/(1/committee) = 18 committees. ‘Republicans’ cancels itself out in the numerator and denominator. We multiply both the numerator and denominator by ‘1 committee’ to move ‘committee’ into the numerator.

We don’t have to see how many committees the Democrats can form. This is because the ratio of Republicans to Democrats in each committee must be 3/2. The ratio of Republicans to Democrats in the Senate is 54/44, which reduces to 27/22. The ratio 3/2 is equal to 33/22, and 33/22 > 27/22, which is the ratio of Republicans to Democrats in the Senate. Since the ratio of Republicans to Democrats needed when forming committees (3/2 = 33/22) is greater than the ratio of Republicans to Democrats in the Senate (54/44 = 27/22), we will run out of Republicans before we run out of Democrats as we form committees.

Just to do the math, the number of committees that could be filled by Democrats is 44/2 = 22, but we run out of Republicans after 18 committees have been formed.

So, the number of committees that can be formed is 18, which is the number of times that 3 (the number of Republicans needed on each committee) can divide evenly into 54 (the number of Republican in the Senate).