In this page set word problem5 we will see how to solve the given
problem using venn diagram.

Using this same method, student can try to solve the given practice problem

100 students were interviewed: 28 took chemistry, 32 took Biology, 40 took Physics, 9 took Chemistry and Biology, 10 took Chemistry and Physics, 8 took Biology and Physics and 4 took all three.

- How many students took none of the three subjects?
- How many students took Chemistry and Biology but not Physics?

Solution:

First let us write the given information.

Total number of students **U** = 100

Number of students took Chemistry **C** = 28

Number of students took Biology ** B** = 32

Number of students took Physics ** P** = 40

Number of students took both Chemistry and Biology = 9

Number of students took both chemistry and Physics = 10

Number of students took both Biology and Physics = 8

Number of students took all three subjects = 4

Let us enter all the details in the venn diagram

We have to enter first the number of students who took all three, that is 4. After that we enter number of students took both Biology and Physics, but in the common area of both the circles we already entered 4, so we have to subtract 4 from 8 and enter the answer 4 in the remaining area common to both B and P only. Similarly we can enter the number of students who took both C and P, and C and B. |

Now we have to enter the number of students who took Physics, 40. But we already entered 14 (6+4+4=14) in P circle. So we have to subtract 14 from 40 and enter the answer 26 in the remaining area of the circle. Similarly we have to enter 19 in B circle and 13 in C circle. |

Now to answer the question how many students took none of the subjects, we have to add all the entries inside the circle and subtract that from the total. 13+5+4+6+4+19+26=77 is the number of students who took either or one of the subject. So who took none of the subject =100-77 =23.

Answer:

- Number of students who took none =
**23** - Number of students who took Chemistry and Biology but not Physics =
**5**

Practice problem: In a class of 175 students 87 registered for Math, 75 registered for English, 60 registered for Science, 28 registered for Math and English, 20 registered for Math and Science, 25 registered for English and Science and 12 registered for all three subjects.

- How many students took none of the subjects?
- How many students took English and Science not Math?

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