Asurvey of 212 spc students was taken at registration. of those surveyed: 74 students had signed up for a math course 51 students had signed up for a language arts course 20 students had signed up for both a math and language arts course 10 students had signed up for both a math and humanities course 5 students had signed up for both a language arts and humanities course 3 students had signed up for all three courses 20 students did not sign up for any of these classes how many students signed up for only humanities (of these three)?

There are 87 students signed up for Humanities course only

Step-by-step explanation:

* Lets explain how to solve the problem

- Its easy to solve this problem using Venn-diagram

- We start with drawing a rectangle and three intersected circles

 inscribed in the rectangle

- The rectangle represents all the students courses

- Circle labeled M for mathematics course

- Circle labeled L for Language Arts course

- Circle labeled H for Humanities course

- There are 74 students had signed up for a Math course

- 51 students had signed up for a Language Arts course

- 20 students had signed up for both a Math and Language Arts course

- 10 students had signed up for both a Math and Humanities course

- 5 students had signed up for both a Language Arts and Humanities

 course

- 3 students had signed up for all three courses

- 20 students did not sign up for any of these classes

- x students had signed up for Humanities course only

* Lets put these numbers in the venn-diagram

∵ There are 20 students did not sign up for any of these classes

∴ Put 20 inside the rectangle but outside the circles

∵ There are 3 students had signed up for all three courses

∴ Put 3 in the common region between the three circles

∵ There are 5 students had signed up for both a Language Arts and

   Humanities course

- 3 student of them are already put in the common part of the 3 circles

∴ Put 5 - 3 = 2 in the common part between circles L and H only

∵ There are 10 students had signed up for both a Math and

   Humanities course

- 3 student of them are already put in the common part of the 3 circles

∴ There are 10 - 3 = 7 in the common part between circles M and

   H only

∵ There are 20 students had signed up for both a Math and

   Language Arts course

- 3 student of them are already put in the common part of the 3 circles

∴ Put 20 - 3 = 17 in the common par between the circles M and L only

∵ 51 students had signed up for a Language Arts course

- We already put 3 + 17 + 2  = 22

∴ Put 51 - 22 = 29 in the empty part of circle L not intersected

  with any other circle

∵ There are 74 students had signed up for a Math course

- We already put 3 + 17 + 7  = 27

∴ Put 74 - 27 = 47 in the empty part of circle M not intersected

  with any other circle

∵ There are x students had signed up for Humanities course only

∴ Put x in the empty part of circle H not intersected with any other

  circle

* Now lets add all the numbers inside the diagram (Bold number)

 and then equate them by the total number of the students

∵ The sum of the students in the venn-diagram is:

  20 + 3 + 2 + 7 + 17 + 29 + 47 + x = 125 + x

∵ There are 212 SPC students was taken at registration

∴ 125 + x = 212

- Subtract 125 from both sides

∴ x = 87

∵ x represents the students had signed up for Humanities

  course only

∴ There are 87 students signed up for Humanities course only

* Look to the attached file for more understand

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