Use of Venn diagrams: Logical Questions

[Qwertyuiop[]]

Hi, I have two logical questions Q1 and Q2. I was able to solve the first one Q1 using venn diagrams, i got 3 variables that i solved by substitution. I don't remember what the answer was so I didn't include it in the diagram but it was correct. If you had to do it, would you solve it by simultaneous equations too or is there a faster to way to do it? I think I have like 2-3 minute for each of these questions, not sure if creating and solving equations is the best method.
For Q2 , there are 2 teams, Football and Skiing. Some pupils of a class either do one sport only, both or neither. To visualize the given information I made a venn diagram but got totally confused at the 5 options. How do I even make sense of the statements? They all sound wrong to me : P .

 

[JeffM]

With respect to the first problem, it is a simple counting question.

n(set A or B) = n(set A) + n(set B) - n(set A and B)

[math]114 + 156 - x = 200 \implies x = 114 + 156 - 200 = 70.[/math]
So 70 wear both. How many wear a waistcoat? 114.

How many wear only a waistcoat? 114 - 70 = 44.

I admit if the number of sets gets larger, Venn diagrams are helpful. But this specific problem can be done much more simply.

For the second problem, I do not think Venn diagrams are much help. I’d try to create counter-examples by being concrete.

Bob is the worst footballer on the ski team. Why does that mean Jack can’t be the best footballer.

 

[Dr.Peterson]

What I'd do is equivalent to your equations, but feels simpler: clearly z=200-114=86, so y=156-86=70, and therefore x=114-70=44.

And I might not use any variables, but just write numbers on the diagram:

​

The second one is a very different kind of problem. All I can suggest is to write out your thoughts about each option; or at least pick the one or two that look like they could possibly be true, and tell us about those. Your diagram might have a little use just to picture where each person asked about would be, but not much more than that.

 

[Steven G]

x+ y = 114
y+ z = 156
x+2y + z =270

x+2y+z = (x+y+z) + y = (200) + y = 270. So y =70

Then x+y = x+70 =114. So x =44

 

[BigBeachBanana]

Hi, I have two logical questions Q1 and Q2. I was able to solve the first one Q1 using venn diagrams, i got 3 variables that i solved by substitution. I don't remember what the answer was so I didn't include it in the diagram but it was correct. If you had to do it, would you solve it by simultaneous equations too or is there a faster to way to do it? I think I have like 2-3 minute for each of these questions, not sure if creating and solving equations is the best method.

For Q1) you'd only need two equations.
[math]\begin{cases} \cancel{x+y=114} \\ y+z=156 \\ x+y+z =200 \end{cases}[/math]Substitute second into third, and solve for x.
[imath]x+y+z =x+156=200 \implies x= 44[/imath]

PS: Please post one question per thread. Mixing the discussions of 2 questions in one thread potentially creates more confusion than it needs to be. If needed, you can provide a link for reference to another thread that you feel is relevant.

 

[Qwertyuiop[]]

For Q2, There 2 or 3 options that I can eliminate. The question says which one is certainly TRUE. (a) and (e) seem wrong to me. So I have (b) (c) and (d) left. Just to remind everyone , (c) is the correct option.

For (a) : The worst footballer among the skiers is the best skier among the footballers.
This one could be right or wrong, we can't know for sure. What if the worst footballer among the skiers is the worst skier among the footballers?

For (e) : If the best footballer doesn't ski, then the best skier doesn't play football. We have no way of knowing if this statement is true. could be true or false. IMO : )

I think we can reject (b) for the same reasons as for the above one. Not a certainly true statement.

Now only (c) and (d) are left.
I noticed in these kind of logic questions, we have to look for the factual statement which is the answer. So it's either (c) or (d) that is the factual statement. What do you think?

 

[Dr.Peterson]

For (a) : The worst footballer among the skiers is the best skier among the footballers.
This one could be right or wrong, we can't know for sure. What if the worst footballer among the skiers is the worst skier among the footballers?

For (e) : If the best footballer doesn't ski, then the best skier doesn't play football. We have no way of knowing if this statement is true. could be true or false. IMO : )

I think we can reject (b) for the same reasons as for the above one. Not a certainly true statement.

I agree; more specifically, who's best or worst in one sport has no correlation with the other sport, and you can easily imagine (but you should actually try to do so!) a contrary case. In (a), you are looking within the overlapping region; in (b) and (e), you are considering someone in each non-overlapping region; but in all three, you are comparing them based on different skills. I'd just ask myself in each case, why couldn't ...?

Now, (c) and (d) are similar to (a), both considering only the overlap; that is, "the footballers who are on the skiing team" are the exact same group as "the skiers who are on the football team". But (c) compares them both on the same basis (age), while (d) compares them on different bases (the best at two different things). Do you see the difference?

I noticed in these kind of logic questions, we have to look for the factual statement which is the answer. So it's either (c) or (d) that is the factual statement. What do you think?

The key idea is whether something must be true; and a good way to test that is to try to imagine a counterexample. If you can (thoughtfully) imagine it not being true, than it isn't necessarily true, and can be eliminated.