Question 34448: In the Junior Class the Followin statistics are true...
*116 students take algebra II
*99 students take Chemistry
*49 take Spanish
*15 students that take algebra II also take Chemistry
*20 students that take algebra II also take Spanish
*22 students that take Chemistry also take Spanish
*12 students take Chemistry algebra II and Spanish
*31 students take none of th three classes
How many students are in the class?
I know the answer is 250... i solved it using a venn diagram and the teacher said it was right... but he said that there is a formulaic way to solve it and is giving MAJOR bonus points if i figure it out... so PLEASE i nned it solved out mathatmatically (meaning a formula)
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! SEE MY WORKING BELOW...
I AM USING * SYMBOL FOR INTERSECTION...X*Y MEANS X INTERSECTION Y.
AND # SYMBOL FOR UNION .....X#Y MEANS X UNION Y.
AND | MEANS COMPLEMENT OR NEGATION.....X| MEANS NOT X.
In the Junior Class the Followin statistics are true...
*116 students take algebra II...=A SAY
*99 students take Chemistry.....=C SAY
*49 take Spanish................=S SAY
*15 students that take algebra II also take Chemistry...= A AND C = A*C
*20 students that take algebra II also take Spanish.....= A AND S = A*S
*22 students that take Chemistry also take Spanish......= C AND S = C*S
*12 students take Chemistry algebra II and Spanish......= C AND A AND S = C*A*S
*31 students take none of th three classes..............= NEITHER C NOR A NOR S = (C|)#(A|)#(S|)
How many students are in the class?
I know the answer is 250... i solved it using a venn diagram and the teacher said it was right...
NOW LOOK AT YOUR VENN DIAGRAM AND ANALYSE....YOU WILL FIND...
A INCLUDES FOUR DISJOINT OR SEPERATE PARTS.. A*C ONLY,A*S ONLY, A ONLY AND FINALLY A AND C AND S
SIMILARLY
C INCLUDES FOUR DISJOINT OR SEPERATE PARTS.. C*A ONLY,C*S ONLY,C ONLY AND FINALLY C AND A AND S
SIMILARLY
S INCLUDES FOUR DISJOINT OR SEPERATE PARTS.. S*A ONLY,S*C ONLY,S ONLY AND
FINALLY S AND A AND C.
HENCE IF WE TOTAL A,C AND S WE GET A#C#S=TOTAL NUMBER OF STUDENTS TAKING ANY ONE OR MORE OF THESE 3 TOPICS BUT INCUDING THE DUPLICATED COUNTS OF THOSE TAKING MORE THAN ONE TOPIC...FURTHER WE FIND WITHIN THOSE ARE THERE TAKING ONLY ONE TOPIC,ONLY 2 TOPICS AND ALL 3 TOPICS..THAT IS MATHEMATICALLY SPEAKING
A INCUDES A*C AND A*C INCLUDES A*C*S...SIMILAR LOGIC HOLDS FOR ALL OTHER PARAMETERS...NOW WITH THE HELP OF VENN DIAGRAM ,WE CAN REMOVE THIS DUPLICATION..METHODICLLY AS FOLLOWS
TOTAL NUMBER OF STUDENTS TAKING ANY ONE OF THE 3 TOPICS(T)
T=(A#C#S)-{A*C+A*S+C*S-(A*C*S)}=A#C#S-A*C-A*S-C*S+A*C*S
TO GET THE TOTAL NUMBER OF STUDENTS IN THE CLASS (G.T)WE SHOULD ADD TO THE ABOVE THE NUMBER OF STUDENTS TAKING NONE OF THE ABOVE 3 TOPICS..THAT IS
G.T=A#C#S-A*C-A*S-C*S+A*C*S+A|#C|#S|
=116+99+49-15-20-22+12+31=250
THAT IS YOUR MATHEMATICAL FORMULA,,
HOPE YOU UNDERSTOOD AND YOU WILL GET YOUR BONUS SCORE...DONT FORGET TO REPLACE THE ABOVE SYMBOLS FOR INTERSECTION(*),UNION(#) AND NEGATION (|)WITH PROPER MATHS
SYMBOLS
but he said that there is a formulaic way to solve it and is giving MAJOR bonus points if i figure it out... so PLEASE i nned it solved out mathatmatically (meaning a formula)
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