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?
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.
I need answer
16 but I still got -46
And:16 but I am still getting -46,39,29,-38,-29
(1) Sorry, I can't draw a Venn diagram for you as I am a text-based bot. But I can describe it to you:
Let's label the circles as follows:
- Chess: 72 students
- Scrabble: 65 students
- Dice: 53 students
To find the number of students playing all three games, first, let's find the number of students playing two games:
- Chess and Scrabble: 35 students
- Chess and Dice: 30 students
- Scrabble and Dice: 21 students
Now, let's find the number of students playing only one game:
- Only Chess: Total Chess players - (Chess and Scrabble + Chess and Dice)
- Only Scrabble: Total Scrabble players - (Chess and Scrabble + Scrabble and Dice)
- Only Dice: Total Dice players - (Chess and Dice + Scrabble and Dice)
(2) To find the number of students playing all three games, we need to subtract the total number of players playing only one game from the total number of students:
Total students - (Only Chess + Only Scrabble + Only Dice)
(3) To find the number of students playing chess alone, we can subtract the number of players in the other two games, who also play chess, from the total number of chess players:
Total Chess players - (Chess and Scrabble + Chess and Dice)
To answer these questions, let's break down the information given using a Venn diagram.
(1) Venn Diagram:
First, we'll draw a Venn diagram with three circles representing Chess, Scrabble, and Dice. We'll label the circles as C, S, and D, respectively.
Next, we'll place the numbers given in the appropriate regions:
- 72 play chess (C)
- 65 play Scrabble (S)
- 53 play dice (D)
- 35 play both chess and Scrabble (C∩S)
- 30 play both chess and dice (C∩D)
- 21 play both Scrabble and dice (S∩D)
- Each play at least one game, which means the total number of students is 120.
This is how the Venn diagram looks like when we include all the given information.
```
C
/\
/ \
30 35
/ \ /
/ X
D -------- S
\ /
\ /
\ /
21
```
(2) Number of Students Playing All Three Games:
To find the number of students playing all three games, we look at the intersecting region inside the circles labeled C, S, and D. From the diagram, we see that this region has the value "X." Therefore, "X" represents the number of students playing all three games.
(3) Number of Students Playing Chess Alone:
To find the number of students who play chess alone, we need to subtract the number of students playing chess with other games (C∩S and C∩D) from the total number of students playing chess (C).
So, the number of students playing chess alone is calculated as:
C - (C∩S + C∩D)
Substituting the given values:
72 - (35 + 30) = 72 - 65 = 7
Therefore, there are 7 students who play chess alone.
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.