Difference in ratio in currency problems and other problems

Question

I know this is very basic stuff, but this is what I am troubling to understand, spend almost 3 hours thinking on that, but can't get clear understanding. I am using 3 examples to put my question here-

1. Let, there are 3 girls per Boy, in a class . Which we can write in ratio G:B = 3:1, in equation G = 3B (G=girl, and B= boy).
2. Let Marry goes 35 miles in 1 hour, which is 35 miles per hour, Which in ratio, Distance covered: Time taken = 35 : 1, in equation D = 35T (D= distance, T= time taken)
3. However, in case of currency, 100 cents worth 1 dollar, or 100 cents per dollar, but in ratio, Cents: Dollar = 1 : 100, in equation 100C = 1D (If C= cent and D= dollar).

My First question is what actually happening in case of currency? Why ratio is- Cents: Dollar = 1 : 100, instead of Cents: Dollar = 100 : 1?

Second question is, in third example, from the equation (100C = D), we can say that 1 D (dollar) worth 100 C (cent). But, how can we put other equations in word?

Answer 1

The best way to do these calculations systematically is to learn about cancelling units.

Link-only answers here are discouraged, but rather than try to explain it from the beginning, I'll point you to many web resources (explanations, tutorials, videos, images). Just google "unit cancellation" and you get

https://www.google.com/#q=unit+cancellation

For example, with units, the fractions $$ \frac{100 \text{ cents}}{1 \text{ dollar}} $$ and $$ \frac{1 \text{ boy}}{3 \text{ girls}} $$ are each just ways to write the number $1$.

You can then use those fractions (with the words) in equations, turn them upside down, cancel the words where they match.

Real fractions are easier to work with than ratios like 3G:1B.