logical reasoning : Premises and conclusion

Question

  Given below are two premises ( A and B). Four conclusions are drawn from 
 them. Select the code that states validly
 drawn conslusion(s) ( taking the premises individually or jointly)

 Premises:
 (A) Most of the dancers are physically fit
 (B) Most of the singers are dancers.

 Conclusions:
 (a) Most of the singers are physically fit
 (b) Most of the dancers are singers.
 (c) Most of the physically fit persons are dancers.
 (d) Most of the physically fit persons are singers.

 Code:

 (1) d and a
 (2) a and b
 (3) b and c
 (4) c and d`

which one is the right answer any why?

source : http://netexam.pmgurus.com/ugc-net-online-questions.aspx?q=solved-ugcnet-paper--july2018&gid=92&h=1&QID=6094&Qno=6

my workout:

i think none of the answers are correct because some or the singers are physically fit and only some of the dancers are singers

and only some of the physically fit persons are singers/dancers.

but not too sure.

Answer 1

If premise:

50.1% of the dancers are physically fit and 50.1% of the singers are dancers.

Conclusions:

< 50% (a) Most of the singers are physically fit.

> 50% (b) Most of the dancers are singers.

> 50% (c) Most of the physically fit persons are dancers.

< 50% (d) Most of the physically fit persons are singers.

Code: (1) d and a (2) a and b (3) b and c (4) c and d

Answer 2

i think none of the answers are correct because some or the singers are physically fit and only some of the dancers are singers and only some of the physically fit persons are singers/dancers.

You are correct that you cannot infer any of the 4 claims, but your reasoning is incorrect.

The easiest way to show that the claims cannot be inferred is by constructing a concrete counterexample. Consider the Venn diagram below:

The numbers indicate how many people there are for each type, e.g. there are 5 people who are physically fit but who are neither dancers nor singers; there is 1 person who is a singer and a dancer, but not physically fit, etc.

If this is the situation, you will find that both premises A and B are true, but all of the possible conclusions a,b,c, and d are false. So, none of the conclusions can be inferred.