in a class of 80 students,40 study physics, 48 study mathematics and 4 4 study chemistry. 20 study physics and mathematics. 24 study physics and chemistry and 32 study only two of the three subjects. if every student studies at least one of the three subjects, find
1 . the number of students who study all the three subjects.
2 . the number of students who study only mathematics and chemistry?
I need solution for the question.
use calculator much faster than using this forum
unless you're trying to figure out how to do somehting
i think icant do it
1. Well, let's put on our thinking caps and solve this puzzle step by step. First, let's find the number of students who study all three subjects. We know that 32 students study only two subjects, so let's subtract that from the total number of students who study both physics and chemistry (24). That leaves us with 24 - 32 = -8. Uh oh, we can't have negative students, so it seems like we made a mistake somewhere. Let's try again!
2. Now, let's tackle the number of students who study only mathematics and chemistry. We know that 4 students study chemistry, and we also know that 32 students study only two subjects. So if we subtract those 32 students from the total number of students who study math and chemistry (which is 48), we'll find our answer. 48 - 32 - 4 = 12. So, there are 12 students who study only mathematics and chemistry.
To find the answers to these questions, we can use the principle of inclusion-exclusion, which allows us to count the number of elements in different sets.
1. To find the number of students who study all three subjects (physics, mathematics, and chemistry), we need to count the number of students who study exactly three subjects.
Let's break down the given information:
- The total number of students in the class is 80.
- 40 students study physics.
- 48 students study mathematics.
- 44 students study chemistry.
- 20 students study physics and mathematics.
- 24 students study physics and chemistry.
- 32 students study only two of the three subjects.
To calculate the number of students who study all three subjects, we can subtract the number of students who study only two subjects from the total number of students who study each subject individually.
Number of students who study all three subjects = Total number of students who study physics + Total number of students who study mathematics + Total number of students who study chemistry - 2 * (Number of students who study only two subjects) - Number of students who study only one subject
From the given information:
- Number of students who study only one subject = 40 (physics) + 48 (mathematics) + 44 (chemistry) - 20 (physics and mathematics) - 24 (physics and chemistry) - 32 (students who study only two of the three subjects)
- Number of students who study all three subjects = 40 + 48 + 44 - 2 * 32 - (40 + 48 + 44 - 20 - 24 - 32)
Now, let's calculate:
Number of students who study all three subjects = 40 + 48 + 44 - 2 * 32 - (40 + 48 + 44 - 20 - 24 - 32)
= 132 - 24 - (156 - 20 - 24 - 32)
= 132 - 24 - 156 + 20 + 24 + 32
= -24
The result is -24. However, it is not possible to have a negative number of students studying all three subjects. Therefore, there must be an error in the given data or calculation.
2. To find the number of students who study only mathematics and chemistry, we need to subtract the number of students who study mathematics, chemistry, or both, from the total number of students who study mathematics.
Number of students who study only mathematics and chemistry = Total number of students who study mathematics - (Number of students who study mathematics and chemistry + Number of students who study only chemistry)
From the given information:
- Number of students who study only mathematics and chemistry = 48 (mathematics) - (20 (physics and mathematics) + Number of students who study only chemistry)
Now, let's calculate the number of students who study only chemistry:
Number of students who study only chemistry = Total number of students who study chemistry - (Number of students who study mathematics and chemistry + Number of students who study only physics and chemistry + Number of students who study only chemistry)
From the given information:
- Number of students who study only chemistry = 44 (chemistry) - (24 (physics and chemistry) + Number of students who study only two of the three subjects + Number of students who study only physics and chemistry)
Now, let's calculate the number of students who study only mathematics and chemistry:
Number of students who study only mathematics and chemistry = 48 - (20 + (44 - (24 + 32)))
Now, let's calculate:
Number of students who study only mathematics and chemistry = 48 - (20 + (44 - (24 + 32)))
= 48 - (20 + (44 - 56))
= 48 - (20 + (-12))
= 48 - 8
= 40
Therefore, the number of students who study only mathematics and chemistry is 40.
If you are studying this topic , you must have learned about Venn diagrams.
Draw 3 overlapping circles.
label the intersection of all 3 as x
low look at the double courses
e.g. 20 study physics and math. That would be the intersection of
the physics and the math circles, which already contains an x
So label the part of physics and math but not chemistry as 20-x
continue likewise for the rest of the data.
You should have all the regions filled in.
Answers can now be written by simply looking at these regions.