Okay, I'll give it a go too :) I'm going to describe it without cross-multiplying. There are many ways to get the answer, and while at first you learn rote methods, gradually you come to understand what you're doing and apply your principles rather than following a set series of instructions blindly :)
You know that the top of the fraction is the numerator and the bottom is the denominator?
If the denominators of two fractions are the same and you are adding or subtracting them, you add/subtract the numberators and put them on top of the denominator, e.g.
3/7 + 2/7 = (3 + 2)/7 = 5/7
4/13 - 1/13 = (4 - 1)/13 = 3/13
So what you need to do to add or subtract any kind of fraction is get them with the same denominator.
In your sum, the three denominators are 2, 5 and 3. So you need to get them all with the same denominator.
Now, Anabel mentioned 'the clever form of 1'. I've never heard it described like that, but as you know, any number multiplied by one is the same number, and 3/3 or 5/5/ or 134/134 = 1. So you can multiply both sides of the equation by 1 and it will stay the same.
That means that 2/3 is the same as 10/15, as I'm sure you've learned when simplifying fractions. It's just the same process in reverse.
What's the 'lowest common multiple' of 2, 5 and 3? In other words, what's the lowest number that has 2, 5 and 3 as fractions? In this case, it's 2 x 3 x 5 = 30, but sometimes it might be lower - but multiplying them together will always work, you just may need to do some more simplifying/cancellnig at the end :)
Now we know that the lowest common multiple is 30, so they all need a denominator of 30. The point of using that number is that you end up with whole numbers on the top half of the fraction, as is correct and appropriate.
To change n/2 to something/30, we multiply by 15/15, which is multiplying by 1 so we don't change anything at all really:
n/2 x 15/15 = 15n/30
Then you need to do the same for n/5 (multiply by 6/6) and 2/3 (multiply by 10/10). That leaves us with:
15n/30 + 6n/30 = 20/30
Note that what we've done here is multiply both sides of the equation by 1, rather than multiplying each part by something different - cos you're not allowed to do that!
Now, everything on either side of the equation has a denominator of thirty. So if you multiplied both sides by thirty, that denominator would disappear and we'd be left with:
15n + 6n = 20
21n = 20
n = 20/21
This method may seem more difficult or longer than Anabel's, but it's the one I instinctively go for, because it's not How To Solve Sums With Fractions, it's tackling it as if it's any maths problem whatsoever :)
If you believe everything you read, better not read.
Japanese Proverb