Strategies:

One strategy is to add together the number of people who like each kind (mytery and fantasy), and subtracting the total number of people who like both.

If I had not said that everyone likes at least one of the types, this would not be the only answer. So if the problem said: There are 60 people my mailing list, and 45 like mysteries, and 39 like fantasy, how many do you think like both? there would have been more than one answer. The answer 24 would be the minimum, and 39 would be the maximum.

One way to explain why this strategy works is to think about direct modeling. If we simplify the problem so that there are only 12 people on the list, 8 like mysteries, and 9 like fantasy, we can make a good model.

Notice that there are 8M's, and 9 F's, so there are 8+9 M's and F's. We can show subtracting 12 by crossing out a letter in each collumn. This leaves us with a letter in each of the collumns that had two letters in before, which are the people in the intersection (overlap)

So, #mystery readers + # fantasy readers - total # readers = # of readers of both mystery and fantasy.

Another way to explain this same strategy is with algebra. For this explanation, I'm going to use the variables:

m = # people who like mysteries but not fantasy; f = # of people who like fantasy but not mystery; b = # of people who like both.

Then m+b is the number off people who like mysteries, and f+b is the number of people who like fantasy. The total number of people is m+b+f. If you add the number of people who like mysteries to the number of people who like fantasy and subtract the total number of people, I can show that with algebra like this:

(m+b) + (f+b) - (m+b+f)

When I simplify this using algebra, I get:

m + b + f + b - m - b - f = m + b + f + b - m - b - f = b

This shows that the calculation gives the number of people who like both.

Finally, someone suggested yet another strategy: If you subtract the number of people who like mysteries from the total number of people, you get the number of people who like fantasy but not mysteries (for the easier version: 12-8=4 people like fantasy but not mysteries). If you subtract the numbe of people who like fantasy from the total number of people, you get the number of people who like mysteries, but not fantasy (12-9=3 people like mysteries but not fantasy). If you subtract those two numbers from the total number of people, you will get the number who like both (12-4-3=5 people like both).

Both of these strategies work just as well for larger numbers as smaller numbers.