I have attached a copy from an old high school textbook that indicates otherwise, without the dollar sign ($) with the 100. See the bottom of the page. I think older textbooks did not use the dollar sign for the calculation, but I do not have any proof to support this belief. It is something that is curious to me, though.
I'd say the book is just ignoring units in the calculation. That's not uncommon; when you calculate, say, E = mc^2, you usually just put in the numbers, knowing that if you used the right units going in, you'll get the right units coming out. It may well be that more recent textbooks try to be more mathematically particular, showing why things are the way they are.
I am also curious to know why does multiplying by 100 or $100.00 represent the finance charge per $100.00 when the denominator is not multiplied by $100.00 as well.
Here is an example: $49.00 represents the finance charge and $200.00 represents the amount financed. The calculation would appear as $49.00/$200.00 * $100.00 = $24.50 the finance charge per $100.00. What I am indicating is that the $200.00 is not multiplied by $100.00 as the $49.00 is, but the result is $24.50 per $100.00, $24.50/$10000. I used a proportion to see how this would appear with $100.00.
$49.00/$200.00? = ?/$100.00
When solving for the unknown amount, the calculation becomes $49.00 * $100.00 = ? * $200.00. After dividing both sides by $200.00 the calculation becomes ($49.00 * $100.00)/$200 = ? This calculation is the one shown above for determining the finance charge per $100.00: $49.00/$200.00 * $100.00. This finance charge per $100.00 calculation appears to be a part of the proportion calculation. I hope you understand what I have explained here.
If you multiply both the numerator and the denominator by the same thing, you're just multiplying by 1 -- the value is left unchanged. Sometimes (as in simplifying a fraction), that's what you want to do; but not in a case like this. You want to get a new number that has a new meaning, not to get back the same number you had. That is, (49*100)/(200*100) is the same number as 49/200, namely 0.245.
On the other hand, in a sense you are multiplying by $100 in both places -- but one of them is remaining as part of the unit!
In your example, you want the
finance charge per $100. In the form you show, you divide $49 by $200 to get a ratio, and then you multiply by $100
per $100! That is, you multiply by $100, and the result is the number you get,
per $100. The denominator is in the unit.
This is identical to what you do in converting a fraction to a percentage. The word "percent" means "per 100"; the 100 is in the name of the unit. When you say 75%, you mean 75/100, or 75 per 100. So when you multiply 3/4 by 100 to get a percentage, you are really multiplying by 100%, that is, 100 per 100 (which is a form of 1, so that the result means the same as the original fraction, but in a different form):
3/4 * 100% = (3/4 * 100)% = 75%
or, in words,
3/4 * 100 per 100 = (3/4 * 100) per 100 = 75 per 100
Now, if you actually "finished" the work by dividing by 100, you'd get 0.75. This is how we convert a percentage back to a decimal or a fraction. Then you have just a number, not a percentage. That is, when we say 75%, we mean 75/100, but we've left that division undone, because it is the meaning of the unit, "%".
Note that your finance charge is, in fact, 0.245 (per dollar), which is 25.5%: 25.5 per hundred.