Of the 600 residents of Clermontville, 35% watch the television show Island Survival,
40% watch Lovelost Lawyers and 50% watch Medical Emergency. If all residents
watch at least one of these three shows and 18% watch exactly 2 of these shows, then
how many Clermontville residents watch all of the shows?
I solved it within 30 seconds. Here is the calculation I did:
100% = 35% + 40% + 50% - 18% - 2x% (x denotes the percentage of people who watch all 3 shows)
So, number of people who watch all 3 shows = 3.5% of 600 = 21, which is the answer.
Now, lets see why I did this way:
We are given percentages for all the categories of people (who watch one show, who watch two shows etc), so it is easier to deal in percentages only. If we are given some amounts in numbers, some in percentages, then we have to convert all these amounts into numbers (or percentages, if one is comfortable).
We are given that all people watch at least one of the shows. So, 100% people watch at least one of the shows.
Now, lets start counting:
35% watch the television show Island Survival
40% watch Lovelost Lawyers
50% watch Medical Emergency
So, adding all these, we get to 125%. So, 125% people watch these shows. But we are given 100% people watch these shows. (in any case, it could not be more than 100% but it can be less) So, we must have double counted people. Lets see in which all cases can we double count or triple count or count a person multiple times.
If say a person X watches one show only, he would be counted only once, among the people watching that particular show, say Island Survival. We don't have a problem with this. We need to count every person once.
If X watches two shows, he would appear in the counting of both shows. But we want him to be counted only once. What do we do? Suppose 10 people watch two shows Island Survival and Lovelost Lawyers, then we would have counted those people two times (once for each show). So, to eliminate this double counting. We need to subtract this 10 from the sum of the people watching shows, to get at the total number of people watching these shows. Thus, we subtract from the sum, all the people who watch two shows.
For example: If I say that there are 10 people who watch show A and 8 people who watch show B and 3 people watch both show A & B. What are total number of people (who watch any of these shows)?
We have A-10
total number - A + B = 18
Is it? No. Because 3 people who watch both shows have been counted twice.
So, total number of people = 10 + 8 - 3 = 15
Let's verify this number. We know out of 15, 3 (let's say X,Y,Z) people see both shows. So, 12 others remaining, who watch either show A or show B.
10 people see show A. Now, out of these 10, three are X, Y & Z. There are 7 remaining who watch show A. So, out of 12 people above, we take out 7 people here. Now, we have 5 people left to watch show B.
So, people who watch show B = 5 + X,Y & Z = 8
So, our calculations are correct.
Now, we come to third category of people, who watch all three shows. These people will be counted thrice. We want to count them once. So, we subtract 2 times their number from the sum.
Thus, we arrive at the equation I mentioned in the beginning:
100 = 35 + 40 + 50 - 18 - 2x
Remember x is in percentage here. Also, remember that we have used 100% because we are given that everyone watches one of these shows. If we were given that 5% population don't watch any of these shows, we would have used 95% as the sum.
If anyone has any queries, please feel free to ask.
PS: Probably some people may not be comfortable with the percentages as I have used above, they may use numbers instead. The logic holds for numbers also.
If this post is helpful, think of giving KUDOS.