90 members of a sport club plays at least one of the game tennis,football and volleyball,10 plays tennis and football,19 play football and volleyball and 29 plays tennis and volleyball.x people plays all the 3 games,while x people each play only one game how many play football and also how many plays football only
I tried using Venn diagrams.
enter x in the intersection of all three
10 play tennis and football, so enter 10-x in the intersection of tennis and football outside the intersection of all three
19 play football and volleyball, so enter 19-x in the intersection of football and volleyball outside the intersection of all three
29 play tennis and volleyball, so .....
Then you say that x people each play only one game, so I placed x in the part dealing only with that one sport
Now add them all up:
x + (10-x) + (29-x) + (19-x) + x+x+x = 90
x + 58 = 90
x = 32
which makes no sense,
e.g. 10-x would be negative
Something not right here, or else I am not interpreting it correctly.
To find out how many members play football and how many play football only, let's break down the given information step by step.
First, we are told that there are 90 members in total and that all of them play at least one of the three games (tennis, football, and volleyball).
Let's define the following variables:
T = Number of members playing tennis
F = Number of members playing football
V = Number of members playing volleyball
X = Number of members playing all three games
Y = Number of members playing only one game
From the given information:
10 members play tennis and football (T ∩ F = 10)
19 members play football and volleyball (F ∩ V = 19)
29 members play tennis and volleyball (T ∩ V = 29)
X members play all three games (T ∩ F ∩ V = X)
We can use the principle of inclusion-exclusion to find the values of T, F, and V.
The principle states:
|T ∪ F ∪ V| = |T| + |F| + |V| - |T ∩ F| - |F ∩ V| - |T ∩ V| + |T ∩ F ∩ V|
From the given information:
|T ∪ F ∪ V| = 90
|T ∩ F| = 10
|F ∩ V| = 19
|T ∩ V| = 29
|T ∩ F ∩ V| = X (unknown)
Plugging in the values, we get:
90 = T + F + V - 10 - 19 - 29 + X
Simplifying the equation:
90 = T + F + V - 58 + X
Rearranging:
T + F + V + X = 148
So, we have obtained an equation relating T, F, V, and X.
Now, let's consider the members playing only one game:
Y = |T ∪ F ∪ V| - |T ∩ F ∩ V|
Y = 90 - X
Since Y represents the number of members playing only one game, we have:
Y = |T - (T ∩ F) - (T ∩ V)| + |F - (F ∩ T) - (F ∩ V)| + |V - (V ∩ T) - (V ∩ F)|
Y = (T - 10 - 29) + (F - 10 - 19) + (V - 19 - 29)
Y = T + F + V - 87
Substituting the value of T + F + V from our previous equation:
Y = 148 - 87
Y = 61
Therefore, there are 61 members who play only one game (tennis, football, or volleyball).
To find the number of members playing football:
F = |F - (F ∩ T) - (F ∩ V)|
F = F - 10 - 19
To determine the value of F, we need additional information from the problem statement or the values of T and V.
Please provide information about T or V, or any other relationships between the variables, so we can find the value of F.