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# How can i answer this question "In an office complex 220 of 605 employees use gym, 285 use cafe and 150 use neither. Represent this in a venn diagram." Also what is the probability of someone randomly selected uses both facilities?

This is an example of a two circle Venn diagram. Not easiest to explain here but I'll try my best. Since you have two categories, it will have two large circles. Since these are not mutually exclusive (they could use both), the circles will overlap in the middle. You know 150 use neither, so you can write this outside the circles immediately. Now you know you had 605 employees but you wrote 150 outside the circles therefore you now only have 605-150 that can be inside = 455. You had 285 using the cafe and 220 using the gym. Now some of these could be in the overlap. Let's set people only using the cafe = c, people using only the gym = g, and people who use both = b. We know 285 = c + b since 285 people use the cafe (includes the people who only use cafe and people who use both). We know 220 = g + b for the same reasoning as above. Let's add these together (as for simultaneous equations) [let me know if you don't know how to use these]. 505 = c + 2b + g We know 455 = c + g + b (total inside the circles of Venn diagram) Therefore, by subtracting these two we get 50 = b. This means our overlap contains 50 people. Using our previously defined equations: 285 = c + b 220 = g + b we can now work out c and g since we know b. c = 235 g = 170 Therefore, our Venn diagram contains 235 people in the circle labelled Cafe, 170 people in the circle labelled Gym, 50 people in the overlap and 150 outside both circles. Let's add these up to check our answer. 235+170+50+150=605 => Everyone is accounted for. Our answer should be correct. Hope this helps!  Alexander Stronach
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Y=x+5

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