venn diagram: which athletes played which seasons

[xc630]

Hi can someone verify my answers for me and help with another part. Thanks.

Of the 415 girls at a high school, 100 played fall sports, 98 played winter sports, ans 96 played spring sports. 22 played all three sports while 40 played only in fall, 47 only in winter and 33 only in spring. How many girls played fall and winter sports but not a spring sport? How many firls did not play sports in any of the seasons.

For the 1st question I got these equations and solved them

38= a+ b
29= b+c
41 = a + c

I got a =25, b=13, c=16 with a as fall& spring , b as winter & fall, and c as winter and spring

For the second question I am not sure what to do. I would appreciate any help

 

[soroban]

Re: venn diagram

Hello, xc630!

Of the 415 girls at a high school, 100 played fall sports, 98 played winter sports, ans 96 played spring sports.
22 played all three sports while 40 played only in fall, 47 only in winter and 33 only in spring.
(A) How many girls played fall and winter sports but not a spring sport?
(B) How many girls did not play sports in any of the seasons.

For the 1st question I got these equations and solved them: \(\displaystyle \:\begin{array}{ccc}38\:=\:a\,+\,b \\29\:=\:b\,+\,c\\41\:=\:a\,+\,c\end{array}\)
I got: \(\displaystyle a =25,\,b=13,\, c=16\,\) with \(\displaystyle a\) as fall & spring,
\(\displaystyle b\) as winter & fall, and \(\displaystyle c\) as winter and spring. \(\displaystyle \;\) . . . Right!

For the second question I am not sure what to do. I would appreciate any help

I assume you made a Venn diagram and filled in the seven spaces with appropriate numbers.

Inside the three circles, you have a total of:
\(\displaystyle \;\;49\,+\,25\,+\,33\,+\,16\,+\,47\,+\,13\,+\,22\:=\:196\) students who played some sport(s).

Therefore, there are \(\displaystyle 415\,-\,196\:=\:219\) students who played no sports.