Question 734096
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.
{{{drawing(300,300,-8,10,-6,12,
blue(circle(0,0,4.5)), red(circle(0,5,5)),green(circle(4,2,4)),
locate(-2,11,red(R=25)),locate(-2,-4.6,blue(C=22)),
locate(6,-1.5,green(W=16)),locate(1.5,2.5,x)
)}}} I labeled the intersection of all 3 circles {{{x}}} 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 {{{10}}} who run and walk, but {{{x}}} of those also cycle, so the ones who run and walk, but do not cycle are {{{10-x}}}.
With a similar reasoning, we figure out that the number of those who run and cycle but do not walk for exercise purposes is {{{11-x}}}.
Similarly, the number of those who walk and cycle but do not run is {{{8-x}}}.
We can those numbers to the diagram, lie this:
{{{drawing(300,300,-8,10,-6,12,
blue(circle(0,0,4.5)), red(circle(0,5,5)),green(circle(4,2,4)),
locate(-2,11,red(R=25)),locate(-2,-4.6,blue(C=22)),
locate(6,-1.5,green(W=16)),locate(1.5,2.5,x),
locate(-2.5,2.5,11-x), locate(2,5,10-x),
locate(2,0,8-x)
)}}}
Now we can calculate the number of people who just cycle, but do not walk or run for exercise as
{{{22-11-(8-x)=22-11-8+x=3+x}}}
Similarly, we can calculate the number of people whose walk, but do not run or cycle for exercise as
{{{16-10-(8-x)=16-10-8+x=x-2}}}
We could also calculate the number who run, but do not walk or cycle for exercise as
{{{25-11-(10-x)=25-11-10+x=4+x}}}
We could add those numbers to the diagram and have:
{{{drawing(300,300,-8,10,-6,12,
blue(circle(0,0,4.5)), red(circle(0,5,5)),green(circle(4,2,4)),
locate(-2,11,red(R=25)),locate(-2,-4.6,blue(C=22)),
locate(6,-1.5,green(W=16)),locate(1.5,2.5,x),
locate(-2.5,2.5,11-x), locate(2,5,10-x),
locate(2,0,8-x),locate(-1,7,4+x),
locate(-1,-2,3+x),locate(5,2.5,x-2)
)}}}
 
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
{{{25+(3+x)+(8-x)+(x-2)=25+3+8-2+x-x+x=34+x}}}
 
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
{{{x-2>=0}}} --> {{{x>=highlight(2)}}} and
{{{8-x>=0}}} --> {{{x<=highlight(8)}}}
Those are the greatest ({{{x=8}}}) and the least ({{{x=2}}}) possible values for {{{x}}} .
 
3. The total number of people (all of whom dis at least one of the 3 exercise activities) is {{{34+x}}} as found in part 1.
According to the greatest value for {{{x}}} found in part 2, above, the greatest value for {{{34+x}}} is
{{{34+8=highlight(42)}}}
 
4. If {{{34+x=40}}} --> {{{x=40-34}}} --> {{{highlight(x=6)}}}