Confusing reasoning/logical question. PLEASE HELP!!!

Question

 

I really need help on these types of questions. I just don't understand how to do them. I don't even understand what math this is lol (algebra 1,2 or geometry). Please help and please tell me what topic to research to understand more of these. Thank you so much!!!

 

Question: Using the information on the table how many teachers teach math only?

 

# of teachers                                   subjects taught

 

12                                                   at least 1 class of math

 

10                                                   at least 1 class of gym

 

20                                                    at least 1 class of science

 

6                                                      both gym and science but not math

 

5                                                      both math and science but not gym

 

2                                                       gym only

 

1                                                      math, gym, and science

Answer 1

This problem is written to teach the use of Venn Diagrams, but say "hello" to Mathematical logic.  [from Wikipedia] "Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory." 

 

[from the Internet]:  "Set theory is used in almost every discipline including engineering, business, medical and related health sciences, along with the natural sciences. In business operations, it can be applied at every level where intersecting and non-intersecting sets are identified. For example, the sets for warehouse operations and sales operations are both intersected by the inventory set. To improve the cost of goods sold, the solution might be found by examining where inventory intersects both sales and warehouse operations."

 

The AND, OR, and XOR (exclusive OR) operations enable a computer can add, subtract, multiply, access memory, compare results, etc (the Arithmetic-Logic Unit, or ALU, is part of a CPU -- almost everyone knows what that is).  Also, if you ever Google, you realize that you can search for logical combinations of terms.

 

Well, to the problem.  Here are three intersecting circles with the numbers filled in.  Read each line in the table given, add the values within a circle, inside a circle but outside another circle, and inside multiple circles.  Make sure to add and subtract (like a computer would).  Then, start with three blank intersecting circles and see whether you can add and subtract the clues to produce this diagram.

 

This Answers Forum doesn’t allow graphics (because WyzAnt intends it to introduce tutors to students), so draw three intersecting circles (well, they can’t be pretty circles) around these numbers:


         Math                                      Science

                5                                     2

                                     1

 

                                     1

                             5              6
 

                                      8

                                    Gym

 

(note: the Math circle goes around 5,5,1,1.  The Science circle goes around 2,1,1,6. And Gym has 8,6,1,5.
 

 

Answer 2

I believe this topic is Set Theory, and it's in algebra and logic. It deals with sets, or collections of objects.

 

In this problem, it'd be helpful to use a Venn Diagram, with 3 intersecting circles labeled math, science, and gym. You want to fill in the more 'specific' sets first.

 

So in the all-intersecting center for all 3 subjects, you'd have 1 teacher in there. In the Gym-only part of the Gym circle, there are 2. That leaves 10 - 1 - 2 = 7 gym people for the remaining Gym slots. As we go up the list, we see that 6 people taught both science and gym but not math. That means there is 1 person taught both math and gym.

 

Continue with the other intersections, and use subtraction to fill in the spaces. Make sure to add up all the spaces in a circle to make sure they add up to the right number.

 

http://imgur.com/YVSU6yx