Question 1207215
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You don't get any benefit from this if we show you the whole solution....<br>
You draw the diagram with the three intersecting circles; then I will explain how to do the work to allow you to answer the given questions.<br>
Some shorthand for my discussion....<br>
A = rap
B = rock
C = heavy metal<br>
Then, for example...
"ABC" is the number of students who like all three types of music
"AC" is the number who like rap and heavy metal but not rock
"B" is the number who like only rock<br>
(1) ABC = 38 (given)<br>
(2) 83 like both rap and rock.  Since 38 students like all three types of music, the number who like rap and rock but not heavy metal is 83-38 = 45:
AB = 45
Use similar calculations to find AC (number who like rap and heavy metal but not rock) and BC (number who like rock and heavy metal but not rap).<br>
(3) "A" is the number of students who like only rap.  A total of 221 like rap; of those, "AB" like both rap and rock, "AC" like both rap and heavy metal, and "ABC" like all three.  So "A", the number who like only rap, is the total number who like rap, minus the numbers who like both rap and any of the other types:
A = 221 - (AB+AC+ABC)
Use similar calculations to find "B" and "C" -- those who like only rock, or only heavy metal.<br>
(4) You now have "A", "B", and "C" -- numbers who like only one of the three types; "AB", "AC", and "BC" -- numbers who like two of the three types; and "ABC" (given) -- the number who like all three types.  The number who like none of the three types is the total number of students (500), minus the sum of all those numbers:
500 - (A+B+C+AB+AC+BC+ABC)<br>
All the sections of your Venn diagram now have the numbers you need to answer the questions.<br>