SOLUTION: A school offers piano lessons and flute lessons to a group of 50 children. Of these children, 28 attend piano lessons 17 attend flute lessons 12 attend neither piano lessons nor

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Question 935871: A school offers piano lessons and flute lessons to a group of 50 children.
Of these children, 28 attend piano lessons
17 attend flute lessons
12 attend neither piano lessons nor flute lessons.
By drawing a Venn diagram, or otherwise, find the number of children who attend only the
piano lessons.
Found 2 solutions by Theo, Edwin McCravy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
t = total number of children.
p = number of children who take piano lessons.
f = number of children who take flute lessons.
p = 28
f = 17
(p or f) = number of children who take piano or flute.
not (p or f) = number of children who take neither.
not (p or f) = 12
(p or f) = t - not (p or f) = 50 - 12 = 38
(p and f) = number of children who take both flute and piano
(p or f) = p + f - (p and f)
38 = 28 + 17 - (p and f)
combine like terms to get:
38 = 45 - (p and f)
solve for (p and f) to get (p and f) = 45 - 38 = 7
(p and f) = 7
number of children who take both piano and flute lessons is equal to 7.
p DISABLED_event_only= p minus (p and f) = 28 - 7 = 21
f DISABLED_event_only= f minus (p and f) = 17 - 7 = 10
the numbers for the venn diagram now become:
t = p only plus f only plus (p and f) plus not (p or f) which becomes:
t = 21 + 10 + 7 + 12 = 50
you would draw a big square which represents the universe.
inside the universe you would draw 2 circles that overlap.
the left circle would be p
the right circle would be f
the overlapping portion of both circles would be (p and f)
the part of the p circle that doesn't overlap would be p only.
the part of the f circle that doesn't overlap would be f only.
the part of the square that is not within either p or f would be not (p or f).
the venn diagram would look like this:
the label for the big square is u which means universe.
$$$

Answer by Edwin McCravy(20064)   (Show Source): You can put this solution on YOUR website!


Suppose we draw a huge red circle P and a huge blue circle F like the above on
an athletic ball field, and require each of the 50 students to stand in the red
circle if the attend piano lessons, to stand in the blue circle if they attend
flute lessons, to stand in both circles if they attend both, but not to stand
in the red circle if they don't attend piano lessons, and not to stand in the
blue circle if they don't attend flute lessons. 

We let a = the number who attend piano but not flute.
We let b = the number who attend both piano and flute.
We let c = the number who attend flute but not piano. 
We let d = the number who do not attend piano or flute.

A school offers piano lessons and flute lessons to a group of 50 children.
Therefore a+b+c+d=50

28 attend piano lessons
Therefore a+b = 28

17 attend flute lessons
Therefore b+c = 17

12 attend neither piano lessons nor flute lessons.
Therefore d=12.

We have the system:



Substituting 12 for d



gives:



Subtracting the second equation from the first gives c=10
Subtracting the third equation from the first gives a=21
Substituting a=21 in a+b=28 gives b=7

So we have:

find the number of children who attend only the
piano lessons.
That's a = 21, the number standing in the red circle P but not in the
blue circle F.

Edwin
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