Answer:
so we know:
total of 20 students
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
chemistry has a total of 14 students
1 2 3 4 5 6 7 8 9 10 11 12 13 14
8 students are in physics
1 2 3 4 5 6 7 8
3 are in all of them
1 2 3
1) underline the main 3
total of 20 students
<u>1 2 3 </u>4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
chemistry has a total of 14 students
<u>1 2 3</u> 4 5 6 7 8 9 10 11 12 13 14
8 students are in physics
<u>1 2 3 </u>4 5 6 7 8
2) lets bold the physics students, if you are in physics you have to do chem so that means 8 of the chem students also take physics.
total
<u>1 2 3 </u>4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
chemistry
<u>1 2 3</u> 4 5 6 7 8 9 10 11 12 13 14
physics
<u>1 2 3 </u>4 5 6 7 8
3) now we bring biology into the "equation"
off the bat we know 3 are definitely in Biology.
so if 4 study ONLY biology we can cross them off our list see the {}
total
<u>1 2 3 </u>4 5 6 7 8 {9 10 11 12} 13 14 15 16 17 18 19 20
chemistry
<u>1 2 3</u> 4 5 6 7 8 {9 10 11 12} 13 14
physics
<u>1 2 3 </u>4 5 6 7 8
biology
<u>1 2 3 </u>
<u />
4) so we have a total of 20 students, 3 in all of the classes <u>(underlined)</u>, 8 in physics (bolded). If there 14 in chemistry, 4 take ONLY chemistry {}, 8 take chemistry as well (bolded), that leaves two students, so they must belong in biology. lets<em> italicise</em> them.
total
<em><u>1 2 3 </u></em>4 5 6 7 8 {9 10 11 12} <em>13 14</em> 15 16 17 18 19 20
chemistry
<em><u>1 2 3</u></em><em> </em>4 5 6 7 8 {9 10 11 12} <em>13 14</em>
physics
<em><u>1 2 3</u></em><u> </u>4 5 6 7 8
biology
<em><u>1 2 3 </u></em><em> 13 14</em>
<u />
5) out of our total of 20 students we know 14 of the students classes, so the remaining six must be in biology.
<u>1 2 3 </u>4 5 6 7 8 {9 10 11 12} <em>13 14</em> 15 <em>16 17 18 19 20</em>
chemistry
<u>1 2 3</u> 4 5 6 7 8 {9 10 11 12} <em>13 14</em>
physics
<u>1 2 3 </u>4 5 6 7 8
biology
<u>1 2 3 </u><em>13 14</em> 15 <em>16 17 18 19 20</em>
6) so that was our "diagram" so now lets count how many are in chemistry and biology but NOT physics.
<u>1 2 3 </u>4 5 6 7 8 {9 10 11 12} <em>13 14</em> 15 <em>16 17 18 19 20</em>
chemistry
<u>-1 2 3</u> 4 5 6 7 8- {9 10 11 12} <em>13 14</em>
physics
<u>-1 2 3 </u>4 5 6 7 8-
biology
<u>-1 2 3 -</u><em>13 14</em> 15 <em>16 17 18 19 20</em>
there are 14 students in chemistry and 11 in biology, 5 of them are in both classes so 14-5= 9 and 11-5= 6 9+6=16. BUT 8 of those sixteen are also in -physics- so we arent counting them, we put those students in dashes. so lets count the students out of the dashes. so<em> twelve</em> of them are in chemistry and biology but not physics.
7) so how many are in just biology but not the other classes? lets look at the graph.
<u>1 2 3 </u>4 5 6 7 8 {9 10 11 12} <em>13 14</em> 15 <em>16 17 18 19 20</em>
chemistry
<u>1 2 3</u> 4 5 6 7 8 {9 10 11 12} <em>13 14</em>
physics
<u>1 2 3 </u>4 5 6 7 8
biology
<u>1 2 3 </u><em>13 14</em> 15 <em>16 17 18 19 20</em>
in the class of biology students 1 2 3 13 and 14 take other classes that leaves students 15 16 17 18 19 and 20. so 6 students take ONLY biology.
sorry its long but i hope this helps!