How many are exposed to two of the three? (venn diagram problem)

Question

Q: A survey has shown that of 100 people chosen at random, 80 watch TV commercials, 70 read newspaper ads, 40 read magazines. Only 10 do none of these things, and 20 do all three. How many of these people are exposed to exactly two of the three forms of advertising?

Here is my reasoning. I gave up on using the union/intersection terminology because I got nowhere.

We are only considering 90 people (100-10).

The sum 80 + 70 + 40 = 190 represents all people in T + N + M including double counting.

190 - 20 = 170 is all people, including doubles, less those who do all 3 (because we only want the sum of the "petals" not the center)

There are only really 90 people meaning that the difference of 170-90 = 80 must represent the sum of the overlapped sections ("petals").

If this is wrong or you know a more elegant way to solve it please share.

Answer 1

I think there are $60$ who get exactly two forms of advertising. Here's why:

If you let $S$ be the number of people who get ads from exactly one source (three regions in the Venn diagram), and let $D$ be the number of people who get ads from exactly two sources (also three regions in the Venn diagram). Then the sum ($190$) of people who get ads from TV, newspaper, magazines (with overcounting) would be equal to $S+2D+3(20)$. So $S+2D=130$.

Also $S+D=70$ (so that the total number of people is $100$).

Conclusion: $D=60$.

Answer 2

If you actually draw a Venn diagram, letting $x,y$ and $z$ be the cardinality of the double-but-not-triple-counted regions, then it quickly becomes apparent that the required answer $x+y+z$ is $160-100=60.$

The sum 80 + 70 + 40 = 190 represents all people in T + N + M including double counting.

190 - 20 = 170 is all people, including doubles, less those who do all 3 (because we only want the sum of the "petals" not the center)

There are only really 90 people meaning that the difference of 170-90 = 80 must represent the sum of the overlapped sections ("petals").

Your mistake (as I hinted above) is forgetting about the triple counting; here's a correction of your line of reasoning:

“The sum 80 + 70 + 40 = 190 represents all people the counts in T + N + M including the double counting and triple counts.

190 - 20 = 170 is all people, including doubles, less those who do all 3 (because we only want the sum of the "petals" not the center)

There are only really 90 people meaning that the difference of 170 190-90 = 80 100 must represent the sum of the overlapped sections ("petals") represents the extra counts $(x+y+z)+2\times20$. Thus, $x+y+z=60.$”