In a group of 120 students, 72 play chess, 65 play Scrabble and 53 play dice if 35 play both chess and Scrabble, 30 play both chess and dice 21 play both Scrabble and dice and each play at least one of the game.
(1) illustrate the information via ven diagram
(2) how many of the students play all the the three?
(3) how many of the students play chess alone?

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  1. Draw 3 intersecting circles, label them C , S, and D for chess, scrabble and dice.
    - enter x in the intersection of all 3
    - "35 play both chess and Scrabble", so place 35-x in the region shown by the intersection of C and S, outside the region that you marked x
    - place 30-x and 21-x in the same corresponding parts of the diagram.
    Now look at the C circle. You have already entered:
    (30-x) + x + (35-x) or 65 - x
    but you are told that C must total 72 ,so
    the part made up of only C = 72 - (65-x) = 7 + x

    Complete the missing parts of the other circles in the same way, so that all
    parts are filled in terms of x
    Finally (sum of all the parts) = 120, and solve for x
    replace all the parts with the value of x and your mystery is solved.

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  2. I need answer

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  3. 16 but I still got -46

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  4. And:16 but I am still getting -46,39,29,-38,-29

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  5. Total chess players =72
    Total scrabble players =65
    Total dice players =53
    Recall: 35 play both chess and scrabble, 30 play both chess and dice 21 play both scrabble and dice
    Let x be the total number of those played all games, then:
    All who play chess = S+35+x+21 = 65, which means that those who play only Scrabble (S) = 9-x
    All who play dice = D+21+x+30 = 53, which means that those who play only dice (D) = 2-x
    All who play chess = C+35+x+30 = 72, which means that those who play only chess (C) = 7-x. Put back 7-x into the above equation, you will get
    7-x+35+x+30 = 72 which implies that 72 - x + x = 72 (by inspection, x = 0)
    that means 72 + x = 72 + x (x zero value, which means none played all the 3 games)
    If you put x = 0, you will get C= 7 (chess alone), S = 9 (scrabble alone) and D = 2 (dice alone) which will add up nicely.

    Wish I could upload the Venn diagram I drew.

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