A math problem?

Question

12 teacher teach Algebra, Geometry, Calculus. 2 teaach Algeebra, 8 teach geometry, and 6 teach caaaalculus. How many teachers teach both geometry and calculus?

Answer 1

There are lots of answers to this question. I'll list a few here, using A to mean teachers who teach only algebra, G for only geometry, etc., AG for algebra and geometry (but not calculus), etc., and AGC for all three. Then some possibilities are:
A ... 1 ...... 1 ...... 0 ....... 0
AG . 1 ..... 0 ...... 1 ....... 0
AC . 0 ..... 0 ...... 1 ....... 0
G ..... 4 .... 5 ...... 5 ....... 4
GC .. 3 .... 2 ...... 2 ....... 4
C ..... 3 .....3 ...... 3 ....... 2
AGC 0 ..... 1...... 0 ........ 2

and there are lots more.

Answer 2

Since there are 12 total teachers but it appears you have 16 (2+8+6), four must be counted in more than one place. This works if one teaches all 3, one teaches only algebra, 5 teach only geometry, and 3 teach only calculus leaving the other two to teach both geometry and calc (but not algebra) plus the one that teaches all three for a total of three teachers teaching geometry and calculus.

Answer 3

You could draw a Venn diagram, and see that the answer could be 2, 3 or 4. But I don't think there is enough information to say for sure. If you said how many teachers teach all three, that would help.

Answer 4

let x be the number of teachers who teach both geometry and calculus.
N umber of teachers who teach only geometry =8-x
N umber of teachers who teach only calculus =6-x
So x+8-x =6-x+2=12
x=4 answer

Answer 5

I think this problem could have multiple answers the way it is written. For example, there could be four teachers who teach both those classes or there could be only 2.

Answer 6

Well no I have 8 digit phone number !