SOLUTION: Asurvey was taken of a number of people as to which of three forms of exercise, running, R, walking, W or cycling, C, they did to keep fit. Each person used one or more of these ex

Question 734096: Asurvey was taken of a number of people as to which of three forms of exercise, running, R, walking, W or cycling, C, they did to keep fit. Each person used one or more of these exercises. The results are as follows: 25 run, 16 walk and 22 cycle. 10 run an walk , 11 run and cycle , 8 walk and cycle .
Let x be the number take all three forms of exercise.
1/ Show that the number of people that use at least one of form of these exercise is 34+ x.
2/ Find : a/ the greatest and b/ the least value of x
3/Hence or otherwise find the greatest number of people that were in the survey
4/ If a total of 40 people took part in the survey , calculate the value of x .

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Let's start by making overlapping circles labeled with R, C, and W (for Running, Cycling, and Walking respectively. We know that some do 2 activities and some do all 3.
I labeled the intersection of all 3 circles for the number of people who do all 3 activities.
When they say "25 run, 16 walk and 22 cycle" they are giving us the total number of people who do each activity,
but some of the people in those totals also do 1 or 2 of the other 2 activities.
That's why I labeled the whole circles on the outside with those numbers.

Now we have to figure out what number of people are represented by each region in the diagram.
There are who run and walk, but of those also cycle, so the ones who run and walk, but do not cycle are .
With a similar reasoning, we figure out that the number of those who run and cycle but do not walk for exercise purposes is .
Similarly, the number of those who walk and cycle but do not run is .
We can those numbers to the diagram, lie this:

Now we can calculate the number of people who just cycle, but do not walk or run for exercise as

Similarly, we can calculate the number of people whose walk, but do not run or cycle for exercise as

We could also calculate the number who run, but do not walk or cycle for exercise as

We could add those numbers to the diagram and have:

1. The total number of people who belong in at least one of those circles could be calculate many different ways (with the same resul). One of those ways is

2. None of those numbers can be negative, because they are numbers of people.
One or more could be zero, but none can be negative, so
--> and
-->
Those are the greatest () and the least () possible values for .

3. The total number of people (all of whom dis at least one of the 3 exercise activities) is as found in part 1.
According to the greatest value for found in part 2, above, the greatest value for is

4. If --> -->
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