Total number of kids + teachers: T + S = 510

Ratio between students and teachers: S:T = 16:1, meaning that S = 16T.

Use last two equations to get:

16T + T = 510
17T = 510
T = 30 ---- S = 480

Make notations:
K = kindergarten ---1/5 of 480 = 96 and you get 480 - 96 = 384 for the rest
a = first grade
b = second grade
c = third grade
d = fourth grade
e = fifth grade
f = sixth grade

e + f account for 1/3 of 384, so a + b + c + d = 2/3 * 384 = 256.

You also know that a + b = 1/4 * 480 = 120.

Use the last two equations to establish that c + d = 2c = 256 - 120 = 136.
This makes c = 68.

Difference between kindergarten and third grade = 96 - 28.
I'm guessing it's all about the speed with which you work. I took me a while to write this answer, but if I didn't have to explain my notations or write what's going on, I'd be a lot faster. In conclusion: use quick notations for yourself and you've already saved some time.
However, I doubt you'll ever see such a problem in the real thing. This one is actually a system of 6 equations! Note that this is my personal opinion....

Actually there is no need to solve equations if you are good with fractions

This question is about how to use fractions efficiently

So here we go

we know that the total number of students is 480

if 1/5 ---> kindergarden ------------------------> 24/120

1/4 1st and 2 nd ---------------> 30/120

1/3 5th and 6th ---------------------> 4/5/*1/3 4/15 = 32/120

so 17/120 = then number of 3rd graders

Hence ans = 480(24-17)/120 = 28

Why ? 120 -----------> LCM (4,5,3) = 60 since 480 and 3rd and 4th graders are equal its easier to take 120