# Counting

• Mar 23rd 2010, 11:54 AM
mjoshua
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?

• Mar 23rd 2010, 12:13 PM
Hello mjoshua
Quote:

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?