In percent change, why do we divide the difference by the original number? If $a = 90$ and $b = 100$, and we are calculating the percent increase going from $a$ to $b$, we calculate the actual numerical difference $b - a = 10$.
But why divide the difference into the starting value $a$? How does seeing how many times $90$ goes into $10$ give a a percentage of increase?
(Obviously we multiply by $100\%$ for a percentage.)
 A: If you change $a=a\cdot (1)$ to $a\cdot (1+d)$ then $d$ is called the proportionate change in $a$. Since "percent" literally means $1/100,$ the  proportionate change in $a$ is $d=(100d)\times (1/100)=100d$ % .
If $a$ changes to $b$ then $a(1+d)=b$ so $d=(b/a)-1=(b-a)/a.$ 
Examples. (1). Your pay rate has been doubled from $a$ to $b=2a.$ Then $d=1$ . You have a $100$% raise.
(2). Your grade has been amended from $90$ to $99$. This is a $10$% increase. The increase $99-90=9$ is $10$% of your original grade  $(90).$
A: One way to think about this might be to consider a situation in which you don't know either the starting or ending number.  Suppose, for example, I tell you that the price of something goes up \$19.  Is that a big increase, or a small increase?  It depends on what the original price was, of course.  If the original price was \$20, then an increase of \$19 is huge -- the price almost doubled.  On the other hand, if the original price was \$240,000 (let's say you're buying a house) than an increase of \$19 is basically pocket change.
So intuitively, the way we apprehend the size of an increase depends on what the original size was.  Percents allow us to quantify that intuitive sense.  In the first example, a \$19 increase on an original price of \$20 is an increase of 95% (that is, the increase is almost as large as the entire original price).  In the second example, a \$19 increase an an original price of \$240,000 is just 0.0079%.  Dividing the increase by the base amount allows you to measure the increase relative to something -- it sets the "scale", so to speak.
A: You are correct in dividing $b-a$ by $a $ to find the percent increase.
It does not make sense to divide $10$ by $100$ to find the percent increase. We have to divide $10$ by $90$, then multiply by $100$ to get the  percentage increase.
