Expert: Paul Klarreich - 12/2/2007

Question
in a local high school, there are 100 students in grade 11.Each student takes at least one of the three sciences: biology,physics, and chemistry. Of the 50 students taking biology, 14 students also take physics and 32 students also take chemistry. Of the 30 students taking physics, 16 also take chemistry. there are 10 students who take all three sciences.

how many students take only chemistry?

how many students take biology or chemistry, but not physics?

Answer
Questioner:   Anna
Category:  Advanced Math
Private:  No
 
Subject:  Mathematical Reasoning
Question:  in a local high school, there are 100 students in grade 11.Each student takes at least one of the three sciences: biology,physics, and chemistry. Of the 50 students taking biology, 14 students also take physics and 32 students also take chemistry. Of the 30 students taking physics, 16 also take chemistry. there are 10 students who take all three sciences.

how many students take only chemistry?

how many students take biology or chemistry, but not physics?
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Hi, Anna,

Here is your standard scheme:

1. Draw a nice big rectangle.  It represents the school.
2. Draw the 3-ring sign.  3 circles that overlap. Label them B,P,C.  I made 3 rectangles below.  The one made of P's is the P-circle.

3. If you did it right, there are 8 regions in the diagram, as shown below. Each region has a specific meaning, and you label them 1-8 as shown.

For example, region 4 is inside P and B, but outside C.  It represents points (students) in P and B, but not in D. (students taking P, B, not C)
=======================================================
WARNING: USE COURIER FONT TO VIEW THIS
===================================================



+-------------------------------------+
|               PPPPPPPPPPPPPPPPP      |
|    BBBBBBBBBBBPBBBBBB         P      |
|    B          P  4  B    6    P      |
|    B        CCPCCCCCBCCC      P      |
|    B   5    C P  1  B 2C      P     |
|    B        C PPPPPPBPPCPPPPPPP     |
|    B        C   3    B C            |
|    BBBBBBBBBBBBBBBBB   C            |
|             C       7  C            |
|             CCCCCCCCCCCC            |
|                                   8 |
|                                     |
+-------------------------------------+


Now work from the inside out.  ABsolutely, positively.

'1' has 10 (why?)
'2' represents C,P, not B. That's the 16 taking C,P, minus the 10 already in '1'. So '2' has 6.

See how it works now?  Don't forget to figure the ones in region 8.  They take none, but you said they all take at least one, so that should work out to zero.