A group of 72 children completed a survey on what kind of activity they like. The choices were: Chess, Swimming, and Football. Everyone liked at least one activity except 7 kids, who don't like any of these three kinds of activities.
12 children liked Chess and Football but not Swimming,
16 children liked Chess and Swimming but not Football,
8 children liked Swimming and Football but not Chess,
10 children liked Chess only,
40 children liked Swimming,
32 student liked Football.
1) Find the number of kids who liked Swimming and Football.
2) What is the probability that a randomly-chosen child from this group likes either Chess or Swimming but not Football?
3) What is the probability that a randomly-chosen child from this group does not like physically active activities?
To find the number of children who liked each activity, we can use a Venn diagram or a system of equations.
Let's use a system of equations.
Let x represent the number of children who liked Chess only.
Let y represent the number of children who liked Swimming only.
Let z represent the number of children who liked Football only.
Let a represent the number of children who liked Chess and Swimming but not Football.
Let b represent the number of children who liked Chess and Football but not Swimming.
Let c represent the number of children who liked Swimming and Football but not Chess.
Based on the given information, we can set up the following equations:
x + a + b + 10 = 72
y + a + c + 40 = 72
z + b + c + 32 = 72
a + b + c + 7 = 72
Simplifying these equations, we get:
x + a + b = 62
y + a + c = 32
z + b + c = 40
a + b + c = 65
To solve this system of equations, we can substitute variables to eliminate some of them.
From the second equation, we know that y + a + c = 32, so we can substitute (32 - a - c) for y in the third equation:
z + b + c = 40
a + b + c = 65
Now, let's substitute (62 - a - b) for x in the fourth equation:
(62 - a - b) + a + b + c = 65
Simplifying, we get:
62 + c = 65
Subtracting 62 from both sides, we find:
c = 3
Using this value of c, we can substitute it back into the second and third equations to find the values of y and z, respectively.
y + a + 3 = 32
z + b + 3 = 40
Simplifying, we get:
y + a = 29
z + b = 37
Now, let's substitute (29 - a) for y and (37 - b) for z in the first equation:
x + a + b = 62
Simplifying, we get:
x = 62 - a - b
Now, we can substitute these values back into the original equation to solve for a:
x + a + b + 10 = 72
(62 - a - b) + a + b + 10 = 72
Simplifying, we get:
72 - a - b = 72 - 10
a + b = 10
Since we now have two equations (a + b = 10 and c = 3) with two variables, we can solve for a and b.
Subtracting the second equation from the first, we get:
(a + b) - (a + b) = 10 - 3
0 = 7
This means that there is no solution for a + b = 10, which indicates that there may be an error in the given information or there is a contradiction in the data provided.
Please double-check the information and provide an accurate explanation of the situation.
Draw your Venn diagram. The first 4 numbers are easy to fill in.
Now let x be those who liked all three
Let F be those who liked only football
Let S be those who liked only swimming
40 liked swimming, and 16+8+x of those liked something else. So,
24+x+S = 40
Similarly,
20+x+F = 32
F+x+S = 19
Solve that and you have
F=3, S=7, x=9
Questions:
(1) 55
(2) 62/65
(3) 7/72