# Why can't ratios be derived from algebraic expressions?

100 cents is worth 1 dollar, so if I wanted to express it in algebra I will write something like this: $$100C=1D$$, which checks out when I try to calculate for some other D.

For eg: how many cents in 2 dollars. Then, $$2D=200C$$, in essence, expression was multiplied by 2 on both sides.

Now, if I wanted to check what's the ratio of dollar(D) to cents, it will be $$D:C=1:100$$. This makes sense since in a dollar there will be more cents than the dollar as cents is a lower denomination. The above ratio can also be written as $$D/C=1/100$$.

But, if I try to write $$D/C$$ from the algebraic expression it gives me: $$D/C=100/1$$

That's the opposite of what I got from my ratio $$D:C=1:100$$. So, what's happening here? I mean it feels intuitive when we are given 14 daks = 1 jin to write $$DAK/J=1/14$$ as ratio but it's wrong. For the purpose of solving questions I can just memorize it but what if someone asks me to explain it?

• Welcome to MSE. You should choose your tags carefully. What has this to do with linear-algebra? Commented Jun 26, 2021 at 10:56
• My bad removed it Commented Jun 26, 2021 at 11:15
• @JoséCarlosSantos: I believe it is common to confuse linear algebra with linear equations such as $2D = 200C$.
– Joe
Commented Jun 29, 2021 at 11:13

You have to be careful about what the symbols $$C$$ and $$D$$ mean. Is it the value of a cent and a dollar (which gives $$D/C=100$$), or is it the number of cents and dollars needed to pay for a particular thing (which gives $$D/C=1/100$$)? You seem to swap between the two without noticing.
• I am trying to understand what's the relation and difference between the two? As you said, $D/C=100$ then it can also be written as $D=100C$. But if I consider in the number of cents and dollars needed to pay for a particular thing  it switches to $D/C=1/100$. It is because of the definition of ratios? a ratio says how much of one thing there is compared to another thing. Could you explain a little on the latter part? Commented Jun 26, 2021 at 11:37
• If $D$ and $C$ represent the value of a dollar bill and a cent coin, then $D$ is the largest one, as a dollar bill is worth a lot more. If $D$ and $C$ represent how many dollar bills or cent coins you need to pay for, say, a bottle of milk, then $C$ is the largest one, as you need many more cents to pay for the milk than you would need dollars. So you have to decide whether the ratio you want is "How much monetary value is there in a dollar compared to a cent", or "How many dollar bills do I need compared to cent coins". Commented Jun 26, 2021 at 12:21