SOLUTION: Can you help me answer this? Thanks In a country then are 100 governors, every governor has 4 assistants. Every governor and assistant serve various committees. Each committee may

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Question 1006385: Can you help me answer this? Thanks
In a country then are 100 governors, every governor has 4 assistants. Every governor and assistant serve various committees. Each committee may be composed of 5 governors, 4 governors and 4 assistants, or 2 governors and 12 assistants. Each governor serves 5 committees, and each assistant serves 3 committees. How many committees are there?
a. 60
b. 80
c. 100
d. 120
e. 160
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is a total of 100%2A4=400 assistants,
and each will serve in exactly 3 committees,
so we will prepare 3%2A400=1200 assistant place cards.
x= number of committees made up of 2 governors and 12 assistants.
y= number of committees made up of 4 governors and 2 assistants.
12x%2B4y=1200 number of assistant place cards prepared for the x%2By committees with assistants.
Since each of the 100 governors serve in exactly 5 committees,
we will prepare 5%2A100=500 governor place cards.
For those place cards to suffice for the committees with assistants, we need
2x%2B4y%3C=500 .
We have the system system%2812x%2B4y%3C=1200%2C2x%2B4y=500%29 ,which graphs as
We need to be in the green-dashed region and on the red line.
If we do not set up any 5-governor committees,
system%2812x%2B4y=1200%2C2x%2B4y=500%29-->system%283x%2By=300%2C2x%2B4y=500%29-->system%28x=70%2Cy=90%29 .
So, forming 70%2B90=160 committees, including
70 committees made up of 2 governors and 12 assistants, and
90 committees made up of 4 governors and 2 assistants,
seems a possibility.
We could rearrange the committees differently, getting rid
Setting up 5-governor meetings sounds like it would lead to less meetings.
However, as shown by common sense , and by 3x%2By=300<-->y=300-3x ,
for every extra 12-assistant committee, we must give up 3 4-assistant committees,
for a total of 3-1=2 less committees with assistants.
using 2 more governor place cards for the extra 12-assistant committee,
and requiring 3%2A4=12 less governor place cards for the 3 eliminated 4-assistant committees.
That would leave us with 12-2=10 governor place cards,
which would be used in 10%2F5=2 5-governor committee.
That way, every time we do that shift,
we eliminate a total of 2 committees with assistants,
but create 2 5-governor committees,
leaving the total number of committees always at highlight%28160%29 .