Question

US Senate Committees The US Senate consists of 100 members. Senate committees are to be formed so that each of the committees contains the same number of senators and each senator is a member of exactly one committee. The committees are to have more than 2 members but fewer than 50 members. There are various ways that these committees can be formed.

a.) What size committees are possible?

b.) How many committees are there for each size?

Answer 1

Essentially, the answer to both these questions rests in the factors of 100. We know that the number of committees and the number of senators in the committee have to be integers. Since there isn't any "double-counting" (senators only belong to one committee), this greatly eases the problem by just allowing us to divide the 100 senators into whatever number of committees to find the number of senators per committee. As such, these two values - the number of committees and the number of senators per committee - will be factors 100 and each pair will multiply to equal 100. All we have to do then is find the factors of 100, which are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The question specifies that the member count in a committee is greater than 2 and less than 50, which means the remaining values for the member counts in the committees are: 4, 5, 10, 20, and 25. All we have to do is pair these values up with an integer so that the product will be 100. These will then be the pairs of what size committees are possible (part a) and how many committees there are for each size.

Answer 2

The members are greater than 2 and less than 50, for each factor you get

4 member committees - 25 of them

5 member committees - 20 of them

10 member committees - 10 of them

20 member committees - 5 of them

25 member committees - 4 of them

Answer 3

right?

Answer 4

ok

Answer 5

i see the factors of 100

Answer 6

by pair you mean multiply to give u 100?

Answer 7

Right the factors of 100 are:
1, 2, 4, 5, 10 , 20, 25, 50, 100

but the question says that the member count for each committee is greater than 2 and less than 50, so the remaining factors are

4, 5, 10, 20 ,25

Answer 8

exactly, they will multiply to give 100 people,

number of committee times number of people in each committee

Answer 9

ok it narrows own the chioces

Answer 10

so by having more than 2 and less than 50 people in each committee, the remaining factors of 100 are,

4,5,10,20,25

Answer 11

if there are 4 people in each committee, there will be 25 commitees

Answer 12

5 people each, then 20 committee

Answer 13

all the factors give you

4 member committees - 25 of them
5 member committees - 20 of them
10 member committees - 10 of them
20 member committees - 5 of them
25 member committees - 4 of them

Answer 14

so part a) is just listing the remaining factors of 100 that are between 2 and 50

4,5,10,20,25 people committee possible

Answer 15

Then part b) for each size , how many committees are there? just 100 divided by each member size.

Answer 16

ok so i do the reverse?

Answer 17

for b?

Answer 18

yeah, for part A) you found that the possible sizes of committees are 4,5,10,20,25 people per committee.

For part B) with 100 total people, how many committees will there be for each of those group sizes. Just take 100 total people, and divide by each of those sizes,
For example, 4 people per committee, there will be 100/4 = 25 committees

Answer 19

divide 100 into 4 for example?

Answer 20

yep, just do that for each of the possible committee sizes.

Answer 21

and you get

4 member committees - 25 of them
5 member committees - 20 of them
10 member committees - 10 of them
20 member committees - 5 of them
25 member committees - 4 of them

Answer 22

i divide 100 into each of the 100's factors to get the how many commitees there is for each size?

Answer 23

cool

Answer 24

second factor 5

5 people per group, 100 total people

100/5 = 20 groups

Answer 25

yeah just divide the total amount of people, by the number of people in the group, to get how many groups there are

Answer 26

possible?

Answer 27

20 people per group....100 total people

\[\frac{ 100 }{ 20 } = 5~~groups ~~of~~ 20 ~~people\]

Answer 28

Just do that for each of the group sizes of: 4,5,10,20,25 and you get

4 member committees - 25 committees
5 member committees - 20 committees
10 member committees - 10 committees
20 member committees - 5 committees
25 member committees - 4 committees

Answer 29

ok

Answer 30

\[\frac{ 100~members }{ 1 }*\frac{ 1~committee }{ 4~members } = \frac{ 100~~committees }{ 4 } = 25~~committees \]