How many girls don’t like either video games or swimming?

Question

attempted to figure this VennDdiagram problem out. I figured one out similar but this one does not work out the same way. Any suggestions to get me started?

Problem: In a group of 14 boys and 14 girls, the number of boys who like video games but don’t like swimming is the same as the number of girls who like swimming but don’t like video games. The number of girls who like video games and swimming is the same as the number of boys who like swimming but don’t like video games. The number of girls who don’t like either video games or swimming is one less than the number of boys who don’t like either video games or swimming. Twice as many girls as boys like video games but no swimming. The number of boys who like video games is one more than the number of girls who like video games. Five boys like video games and swimming. How many girls don’t like either video games or swimming?

Answer 1

Let $a,x$ be the number of girls/boys, respectively, who like only video games.

Let $b,y$ be the number of girls/boys, respectively, who like only swimming.

Let $c,z$ be the number of girls/boys, respectively, who like both activities.

Let $d,w$ be the number of girls/boys, respectively, who like neither activity.

Below is the information we have from the problem.

1. "In a group of 14 boys and 14 girls": $a+b+c+d=x+y+z+w=14$.

2. "the number of boys who like video games but don’t like swimming is the same as the number of girls who like swimming but don’t like video games": $b=x$.

3. "The number of girls who like video games and swimming is the same as the number of boys who like swimming but don’t like video games": $c=y$.

4. "The number of girls who don’t like either video games or swimming is one less than the number of boys who don’t like either video games or swimming": $w-1=d$.

5. "Twice as many girls as boys like video games but no swimming": $a=2x$.

6. "The number of boys who like video games is one more than the number of girls who like video games": $x+z=a+c+1$.

7. "Five boys like video games and swimming": $z=5$.

We want to find $d$.

Subbing in 5, 7 into 6 gives: $x+c=4.$

This, together with 1 and 3 gives: $4+z+w=14\implies z+w=14\implies w=9$.

From 4, we get $d=9-1=8$.

Thus, 8 girls don't like either video games or swimming.