11o members of a sports club play at least one of the games, football basketball and volleyball. If 20 play football and basketball only,15 play football and volleyball only, 26 play basketball and volleyball only, x play all the three game,2x each play only one game, how many play basketball altogether?
better recheck your figures, Reiny. There's no "-x" needed, since it says
20 play football and basketball only
That is, the x in the center does not need to be subtracted from each intersection.
if you draw the Venn diagram, you can easily see that
20+26+15+x+2x+2x+2x = 110
find x, and then you want 20+26+2x+x
I stand corrected, skimmed and stumbled over that "only"
Assumed the usual wording for this type of question
To find out how many members play basketball altogether, we need to analyze the given information step by step.
Let's denote the number of members who play only football as F, the number of members who play only basketball as B, and the number of members who play only volleyball as V.
From the given information, we know that:
- 20 members play football and basketball only (F ∩ B = 20)
- 15 members play football and volleyball only (F ∩ V = 15)
- 26 members play basketball and volleyball only (B ∩ V = 26)
We are also given that x members play all three games. So, the total number of members who play basketball, including those who play other games as well, is given by:
B + (F ∩ B) + (B ∩ V) + x
Additionally, we are told that 2x members each play only one game. So, we have:
(F ∩ B ∩ V) = 2x
Now, let's use this information to determine the number of members who play basketball altogether.
First, let's find the value of x:
From the information we have, the total number of members who play each game only once is:
(F ∩ B ∩ V) = 2x
And, we know that the total number of members who play each game only once is:
(F ∩ B ∩ V) + (F ∩ B) + (F ∩ V) + (B ∩ V) = 2x + 20 + 15 + 26
Since the total number of members in the sports club is 110, we can write an equation:
(F ∩ B ∩ V) + (F ∩ B) + (F ∩ V) + (B ∩ V) + x + (B + F + V) = 110
Substituting the values we have:
2x + 20 + 15 + 26 + x + (B + F + V) = 110
Rearranging the equation:
3x + (B + F + V) = 110 - (20 + 15 +26)
Simplifying:
3x + (B + F + V) = 49
Since the number of members who play only one game is given by:
(B + F + V) = 2x
We can substitute that value back into the equation:
3x + 2x = 49
Simplifying further:
5x = 49
Dividing both sides by 5:
x = 49/5
Since x represents the number of members who play all three games, it cannot be a fraction. So, we need to find a whole number that satisfies the equation. Since x cannot be a fraction, the value 49/5 cannot be the actual value of x. Therefore, we need to choose the nearest whole number as the value of x, which is 10 (approximately 9.8).
Now, let's find the number of members who play basketball altogether:
B + (F ∩ B) + (B ∩ V) + x = B + 20 + 26 + 10
Simplifying:
B + 56 = B + 46
By subtracting B from both sides:
56 = 46
This is not possible and indicates that there may be an error in the given information. Please double-check the figures provided to ensure the accuracy of the problem.
I made my Venn diagram, filled in the data
x in the middle, 20-x, 15-x, and 26-x for the corresponding 2-item sports, and 2x in each of the circles, for students involved in only one sport.
My equation is
2x + x + (20 - x) + (15 - x) + 2x + (26 - x) + 2x = 110
4x = 49
x = 49/4 ,which is not a whole number since x represents number of students
I think your data is flawed.
The solution for oobleck's equation is
x = 7
filling in the Venn diagram with that solution, gives a total of 89 students involved in the 3 sports, not 110 as needed.