How to solve Venn Diagrams

Question

A pollster on a college campus interviewed the women students as to the shampoo they used.  Of the 7900 on campus, she reported that 4500 used pantene, 4850 used suave, and 4750 used garnier fructis.  Of these women, 3000 used both pantene and suave, 2750 used suave and garnier fructis, 2250 used pantene and garnier fructis, and 1750 used all three.  Was this poll accurate? Why or why not?

Answer 1

To use a Venn diagram, we use one large circle to represent the 7900 women. Inside the circle we have 3 interlocking circles representing Pantene users (P), Suave users (S) and Garnier Fructis (F). The interlocking circles generate 7 zones that are the areas enclosed by each interlock. An 8th zone is outside the interlocking circles but inside the big circle.

The 8 zones are (Zn):

1. Using only Garnier Fructis
2. Using only Pantene
3. Using only Suave
4. Using both Garnier Fructis and Pantene, but not Suave
5. Using both Garnier Fructis and Suave, but not Pantene
6. Using both Pantene and Suave, but not Garnier Fructis
7. Using all three (1750)
8. Using none of them

All the given facts can be represented in equations connecting the zones and using the given figures.

F=Z1+Z4+Z5+Z7=4750; P=Z2+Z4+Z6+Z7=4500; S=Z3+Z5+Z6+Z7=4850; Z6+Z7=3000; Z5+Z7=2750; Z4+Z7=2250; Z7=1750. From this Z6=1250; Z5=1000; Z4=500. Z1+...+Z8=7900.

Z1=4750-(500+1000+1750)=4750-3250=1500; Z2=4500-(500+1250+1750)=4500-3500=1000; Z3=4850-(1000+1250+1750)=4850-4000=850; Z8=7900-(1500+1000+850+500+1000+1250+1750)=7900-7850=50.

So the number of women in each zone has been evaluated. Only 50 women out of 7,900 use none of the products.

The poll is accurate and comprehensive because the fact that every zone possible has been evaluated leaves no unknown figures.