A survey o 85 students asked them about the subjects they liked to study 35 students liked maths,37liked history,26 liked physics,20 liked maths and history,14 liked maths and physics,2 liked all the subjects. a) how many of these students like maths and physics? B) how many students did not like any of 3 subjects? c) how many like maths and history but not physics? Solve the problem using ven diagram
To solve this problem using a Venn diagram, we can start by drawing three overlapping circles to represent the subjects: maths, history, and physics.
First, we are given that 35 students liked maths, 37 students liked history, and 26 students liked physics. Let's write these numbers in the relevant sections of the Venn diagram:
- In the circle representing maths, we write 35.
- In the circle representing history, we write 37.
- In the circle representing physics, we write 26.
Next, we are given that 20 students liked both maths and history, 14 students liked both maths and physics, and 2 students liked all three subjects. Let's add these numbers to the overlapping regions:
- In the overlapping region between maths and history, we write 20.
- In the overlapping region between maths and physics, we write 14.
- In the overlapping region in the middle where all three subjects intersect, we write 2.
Now, let's solve the individual parts of the problem using the Venn diagram:
a) How many of these students like maths and physics?
To find this, we add the numbers in the overlapping region between maths and physics. From the diagram, we can see that 14 students liked both maths and physics.
b) How many students did not like any of the three subjects?
To find this, we need to calculate the number of students outside all three circles, which is the complement of the entire diagram. We can do this by subtracting the total number of students from the sum of the numbers in each circle and the intersections:
Total number of students = 85
Number of students who liked maths = 35
Number of students who liked history = 37
Number of students who liked physics = 26
Number of students who liked both maths and history = 20
Number of students who liked both maths and physics = 14
Number of students who liked all three subjects = 2
Number of students who did not like any of the three subjects = Total number of students - (Sum of numbers in each circle and intersections)
Number of students who did not like any of the three subjects = 85 - (35 + 37 + 26 - 20 - 14 - 2)
Number of students who did not like any of the three subjects = 85 - 82
Number of students who did not like any of the three subjects = 3
Therefore, 3 students did not like any of the three subjects.
c) How many like maths and history but not physics?
To find this, we need to subtract the number of students who liked all three subjects from the number of students who liked both maths and history. From the diagram, we can see that 20 students liked both maths and history, and 2 students liked all three subjects.
Number of students who liked maths and history but not physics = Number of students who liked both maths and history - Number of students who liked all three subjects
Number of students who liked maths and history but not physics = 20 - 2
Number of students who liked maths and history but not physics = 18
Therefore, 18 students like maths and history but not physics.
To solve the problem using a Venn diagram, we can visually represent the relationships between the subjects and the number of students who like each subject.
First, draw three overlapping circles to represent the three subjects: math, history, and physics.
Step 1: Fill in the given information:
35 students like math,
37 students like history,
26 students like physics,
20 students like math and history,
14 students like math and physics, and
2 students like all three subjects.
- Fill in the circle for math with 35.
- Fill in the circle for history with 37.
- Fill in the circle for physics with 26.
- Write 20 in the overlapping region of math and history.
- Write 14 in the overlapping region of math and physics.
- Write 2 in the overlapping region where all three subjects overlap.
Now we can calculate the remaining numbers:
a) To find the number of students who like math and physics, we need to add the numbers in the overlapping region of math and physics, which is 14.
b) To find the number of students who do not like any of the three subjects, we need to find the remaining students outside all three circles. To do this, we add up the numbers in the circles (35 + 37 + 26) and subtract the overlapping regions (20 + 14 + 2). So, 35 + 37 + 26 - 20 - 14 - 2 = 62 students do not like any of the three subjects.
c) To find the number of students who like math and history but not physics, we subtract the number in the overlapping region of math, history, and physics (2) from math and history (20). So, 20 - 2 = 18 students like math and history but not physics.
Therefore, the answers to the questions are:
a) 14 students like math and physics.
b) 62 students do not like any of the three subjects.
c) 18 students like math and history but not physics.