.
The condition is:
In a cultural gathering of 400 people, there are 270 men and 200 musicians. (A)
Of the latter, 80 are singers. (B)
60 of the women are not musicians and 220 of the men are not singers. (C)
How many of the women are musicians but not singers,
if there are 150 singers altogether and 40 men are both musicians and singers? (D)
Solution (Abbreviations: cond.A means "condition A"; n.3 means the reference to n.3)
1. Since of 400 people there are 270 men (cond.A), then the number of women is 400 - 270 = 130.
2. Since of 130 women 60 are not musicians (cond.C), then the number of women who ARE musicians is 130 - 60 = 70.
3. Since of 270 men 220 are not singers (cond.C), then the number of men who ARE singers is 270 - 220 = 50.
4. Since there are 150 singers altogether (cond.D) and the number of men who are singers is 50 (n.3), then
the number of women who are singers is 150 - 50 = 100.
5. Since there are 200 musiciants (cond.A) of whom 70 are women (n.2), then the number of men who are musicians is 130.
6. Now let's collect in the Table what we got:
In all Musicians Singers
Men 270 (cond.A) 130 (n.5) 50 (n.3)
Women 130 (n.1) 70 (n.2) 100 (n.4)
Let's continue our analysis.
7. Since there are 200 musicians (cond.A) and of the latter 80 are singers (cond.B), then 80 are musicians AND singers
simultaneously.
8. Since 40 men are both musicians and singers (cond.D), then the number of women who are musicians AND singers
simultaneously is 80 - 40 = 40 (the number 80 came from n.7).
Again, let's summarize this: there are 130 women; of them 70 are musicians, 100 are singers (<<<--- see the Table !)
and 40 women are both musicians AND singers (<<<--- n.8 !)
Having this, everybody can answer the major question: "How many of the women are musicians but not singers ?"
70 - 40 = 30.
Answer. 30 women are musicians but not singers.
S O L V E D ! ! !