Explanation:
To determine the missing values in the Venn diagram, we can use the information from the problem to set up a system of equations. Let's assign variables to each section of the diagram:
X = number of students who play tennis only
Y = number of students who play soccer only
Z = number of students who play volleyball only
W = number of students who play tennis and soccer only
y = number of students who play tennis and volleyball only
Using the information from the problem, we can set up the following equations:
X + W + 3 = 21 (21 students play tennis)
Y + W + 4 = 16 (16 students play soccer)
Z + y + 5 = 19 (19 students play volleyball)
W + 3 = 3 (3 students play soccer and tennis)
y + 5 = 5 (5 students play volleyball and tennis)
Solving for one variable in terms of the others, we can substitute into the other equations:
X = 21 - W - 3
Y = 16 - W - 4
Z = 19 - y - 5
Substituting into the equation for W, we find:
W + 3 = 3
W = 0
Substituting W = 0 into the equations for X and Y:
X = 21 - 0 - 3 = 18
Y = 16 - 0 - 4 = 12
Finally, we can substitute X, Y, and W into the equation for Z to find:
Z = 19 - y - 5
Z = 19 - 5 - 5 = 9
So the missing values in the Venn diagram are:
Soccer: 12 students
Volleyball: 9 students
Tennis: 18 students
Soccer & Tennis: 0 students
Volleyball & Tennis: 5 students
Volleyball & Soccer: 4 students
Note that the number of students who play none of the three sports is not explicitly stated in the problem, but it can be calculated as:
13 = total number of students - (X + Y + Z + W + y)
13 = total number of students - (18 + 12 + 9 + 0 + 5)
13 = total number of students - 44
So there are 57 total middle school students