1. ## Counting

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?

2. 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?

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 $x$ in the overlap. Then work outwards:
There are $9$ altogether in J; we've placed $x$ of them in the overlap, so there must be $9 - x$ left in the other part of the loop.

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

There are $9$ 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 $27$:
$x + (9-x) + (12-x) + 9 = 27$

$\Rightarrow 30 - x = 27$

$\Rightarrow x = 3$
OK now?