Of the 200 employees in a certain company, 25 percent will

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Hi Bunnel,

Will the minimum number of satisfied employees be 30?

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Yes, absolutely.

i know the min/max concept is pretty basic but conceptually can one use overlapping sets/Venn diagram to solve this and other min/max problems? I'm not sure how to arrive at the min value (maybe someone could provide their input) but to solve for the max i think you can set-up two sets of two overlapping sets.

For display purposes i have found a sample overlapping set.



The first set would represent the max value for City X and the second set would be for City Y. Circle A represents the total amount of people relocating to City X and the Circle B represents the total that prefer City X. To maximize the number of people that are relocated to their preferred city you would have to have the highest possible value from Circle A in the overlapping set. So for City X, circle A, the overlapping area and circle B would contain the following amounts: 0,50,120.

For City Y the amounts would be 70,80,0. The combined overlapping sets would equal the answer 130.

It will come up in this way....

The wording is perhaps the only obstacle- the actual math is relatively simple. How? Well, what the question is basically stating is that 25 percent of employees will move to city X and 75 percent of employees will move to city Y. So the boss is pretty much saying ok 50 of you are going to City X and 150 of going to City Y regardless of who wants what. Who wants what? 80 of the employees want to go to city Y and 120 want to go to City X- but that of course cannot happen because only 50 employees will be moved to city X on the boss' order. So Human Resources basically steps in and says well okay we care about employee happiness and we want to minimize the number of unhappy employees so we need to make sure we get as many employees as possible moved to the city they want to work in. From here have to try to create a distribution of employees that maximizes the placement of employees into City's they want to go to.

As we can see from the diagram- we can put all 80 citizens that want to go to city Y into city Y because city Y requires 150 employees on the boss's order. But wait we still need 70 more employees... so okay how about we move 50 employees that want to work at City X to City X and the 70 employees left that don't want to go to City Y...well too bad because we've just put 80 employees that wanted to go City Y at City and 50 that want to go to City X at X so we've made 130 employees happy- so the other 70 can go suck it up.

Given: Of the 200 employees in a certain company, 25 percent will be relocated to City X and the remaining 75 percent will be relocated to City Y. However, 40 percent of the employees prefer City Y and 60 percent prefer City X.
Asked: What is the highest possible number of employees who will be relocated to the city they prefer?

	Relocated to City X	Relocated to City Y	Total
Prefer City X	x	120-x	120
Prefer City Y	50-x	x+30	80
Total	50	150	200

Number of employees relocated to city they prefer = x + x + 30 = 2x + 30
x can be max 50

	Relocated to City X	Relocated to City Y	Total
Prefer City X	50	70	120
Prefer City Y	0	80	80
Total	50	150	200

Max number of employees relocated to city they prefer = 2x + 30 = 130

IMO D

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