Is the Answer 6%?

gmatkkvinu wrote:A, B and C are three TV channels. A survey shows that 30%, 20% and 85% of the people in a locality watch A,B and C respectively, 20% watch exactly two of the three channels and 5% watch none.

If the same survey indicates that 20% watch A and C and 16% watch B & C, then what percent of people watch only channel A?

Since 5% watch none of the shows, the total who watch one or more shows = 95.
Here is one formula for triple-overlapping groups:

T = A + B + C - (AB + AC + BC) - 2(ABC)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in A, everyone in B, and everyone in C:
Those in exactly 2 of the groups (AB+AC+BC) are counted twice, so they need to be subtracted from the total ONCE.
Those in all 3 groups (ABC) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

In the problem above:
Let T = 95.
A = 30.
B = 20.
C = 85.
Exactly 2 of the channels = 20. .
Let ABC = the percentage who watch all 3 channels.

Plugging these values into the formula, we get:
95 = 30 + 20 + 85 - 20 - 2(ABC)
95 = 115 - 2(ABC)
ABC = 10.

To determine the percentage who watch ONLY A, draw a Venn Diagram.
Enter the data from the INSIDE OUT.

A= 0, B=20, C=85.
ABC = 10.


20 watch A and C.
16 watch B and C.
Subtracting from these figures the 10 who watch all 3 channels, we get:


20 watch exactly 2 of the channels.
Since AC=10 and BC=6, AB=4:


Subtracting from A=30 the values for AB, AC, and ABC, we get that Only A = 30 - (10+4+10) = 6:


The correct answer is [spoiler]Only A = 6.[/spoiler].