I'll do problem 6 to get you started
x = number of people training all 3 sports.
This is because all 3 sports are needed for the triathlon
There are 35 swimmers total. 9 of which are also cyclers (but not runners), and there are 13 swimmers who are also runners (but not cyclers)
That leaves 35-9-13-x people who are swimmers only. I'll leave this expression un-simplified so you can see where the numbers are coming from. Take note of where I'm placing this in the venn diagram below.
There are 32 cyclers. Of this group, 9 are swimmers (but not runners) and 11 are runners (but not swimmers). That leaves 32-9-11-x people who are cyclers only.
There are 38 runners. Of this group, 13 are swimmers (but not cyclers) and 11 are cyclers (but not swimmers). That leaves 38-13-11-x people who are runners only.
All of these values are honestly hard to keep track of. This is why a visual approach using a Venn Diagram is recommended. See the diagram below.
The idea is that every value in the diagram then adds up to the 58 total attendees. This helps form the equation needed to solve for x.
Such an equation is possible because each attendee participates in at least one of the sports mentioned. No values will be outside of all three circles at the same time.
(35-9-13-x)+(32-9-11-x)+(38-13-11-x)+(9)+(13)+(x)+(11)=58
-2x+72 = 58
-2x = 58-72
-2x = -14
x = -14/(-2)
x = 7
Answer: There are 7 people training for all three sports
