Hello mjoshua Originally Posted by

**mjoshua** Is there an easier way to do questions like these? (I stuggle with them sooo much and all the overlap/non-overlap).

In a class of 27 students, 9 students take Japanese, 12 students take government, 9 neither take Japanese nor government. How many students take both Japanese and government?

(And the answer is 3).

Easier than what? Guesswork?

The standard way is to draw a Venn diagram like the one that I've attached.

You'll see that I have drawn two overlapping loops representing the students who take Japanese (J) and government (G). Since we don't know how many take both, write $\displaystyle x$ in the overlap. Then work outwards:There are $\displaystyle 9$ altogether in J; we've placed $\displaystyle x$ of them in the overlap, so there must be $\displaystyle 9 - x$ left in the other part of the loop.

In the same way, there are $\displaystyle 12 - x$ in the right-hand part of loop G.

There are $\displaystyle 9$ who aren't in either loop. These go outside the loops, then.

Then add up the numbers in each of the four regions, putting the total equal to $\displaystyle 27$:

$\displaystyle x + (9-x) + (12-x) + 9 = 27$

$\displaystyle \Rightarrow 30 - x = 27$

$\displaystyle \Rightarrow x = 3$

OK now?

Grandad