In a class of 60 pupils, 46 passed English and 42 passed math. Each pupil passed at least one of the subjects. I) illustrate this information on a van diagram. ii) how many pupils passed both subjects? Let's y represent the number of pupils who passed both subjects
To illustrate this information on a Venn diagram, we can start by drawing two circles to represent English and Math.
Inside the English circle, we write 46, representing the number of pupils who passed English. Inside the Math circle, we write 42, representing the number of pupils who passed Math.
Next, we need to include the information that each pupil passed at least one of the subjects. To do this, we draw an overlap section where the two circles intersect. Inside this overlap section, we write y, representing the number of pupils who passed both English and Math.
In the remaining parts of the circles outside the overlap section, we write the numbers of pupils who passed only English or only Math.
The completed Venn diagram would look like this:
English (46)
___________
/ \
/ \
/ \
/ \
/ \
(y) ______________ ______ (42)
English Only Math Only
To find the number of pupils who passed both subjects (y), we need to consider the total number of pupils in the class. In the Venn diagram, we have accounted for the pupils who passed only English (46 - y) and the pupils who passed only Math (42 - y).
According to the given information, each pupil passed at least one subject.
So, we can set up an equation using the total number of pupils in the class:
60 = (46 - y) + (42 - y) + y
Simplifying this equation, we get:
60 = 46 + 42 - y
Combining like terms:
60 = 88 - y
Rearranging to solve for y:
y = 88 - 60
y = 28
Therefore, 28 pupils passed both English and Math.
I) To illustrate this information on a Venn diagram, we can use two overlapping circles. One circle represents English, and the other represents Math.
First, we will fill in the information that 46 pupils passed English and 42 pupils passed Math.
- In the English circle, write "46" to represent the number of students passing English.
- In the Math circle, write "42" to represent the number of students passing Math.
Next, we know that each pupil passed at least one subject. This means that there will be some overlap between the circles. I will label this overlapping region "y" to represent the number of pupils who passed both subjects.
II) To determine how many pupils passed both subjects, we need to find the value of "y".
We know that the total number of students in the class is 60, and each student passed at least one subject.
Using the principle of inclusion-exclusion, we can calculate the number of students who passed both subjects as follows:
Total students passed English + Total students passed Math - Total students passed both subjects = Total students in the class
46 + 42 - y = 60
88 - y = 60
To isolate "y" on one side of the equation, we subtract 88 and add y to both sides:
-y = 60 - 88
-y = -28
Multiply both sides by -1 to get a positive value for "y":
y = 28
Therefore, there are 28 pupils who passed both English and Math.
To illustrate this information on a Venn diagram, you can use two overlapping circles, one representing English and the other representing math.
1. On the Venn diagram, label the circle for English as "English" and the circle for math as "Math."
2. Write the number of pupils who passed English only (46) inside the English circle.
3. Write the number of pupils who passed math only (42) inside the math circle.
4. Since each pupil passed at least one subject, there should be values outside the circles. Write any such value in the region where the two circles overlap, indicating students who passed both subjects.
5. If we represent the number of pupils who passed both subjects as "y," write "y" in the overlapping region of the two circles.
Now, let's move on to determining the value of "y," the number of pupils who passed both subjects.
To find the number of pupils who passed both subjects (y), we need to determine the overlap between the two circles on the Venn diagram.
According to the given information, out of 60 pupils in total, 46 passed English, and 42 passed math.
Since each pupil passed at least one subject, we can add the number of pupils who passed English only and the number of pupils who passed math only to get the total number of pupils who passed at least one subject:
46 + 42 = 88
However, we know that the total number of pupils cannot exceed the total number of pupils in the class, which is 60.
Therefore, we need to use the following equation to find the number of pupils who passed both subjects (y):
(Number of pupils who passed English only) + (Number of pupils who passed math only) - y = Total number of pupils who passed at least one subject
46 + 42 - y = 88
Now we can solve for y:
88 - 46 - 42 = y
y = 0
Therefore, based on the given information, zero pupils passed both English and math.