Why does 3600:6 have the same quotient as 1800:3?

Think of it this way:

3600 divided by 2 is 1800

and 6 divided by 2 is 3.

Since both the dividend and the divisor were divided by 2, the quotient is the same for both because the ratios were the same.

3600 = 1800

6 3

cross multiply

10,800=10,800

The quotients are the same (600).