I think you're a little confused with why 1 and 100% are the same
For that u need to understand the relationship between a fraction value and it's equivalent percentage value
When we say x is (1/2) of y that means x is half the value of y... And it also means x is 50% of the value of y...
When we say something is (1/3) that means it is one third the value... Or in other words we also say it's 33.33% of the value....
So what's the relationship?
The relation is that if something is x%, that means in fraction term it is (x/100)
50% = 50/100 = 1/2
33.33% = 33.33/100 = 1/3 and so on
Similarly if something's value does not change, or in other words x=y, we say x is 100% of y...
So in fraction term it's 100/100 = 1
Now coming to % increase
For that we first understand increase and then move on to increase %
When a value changes from 10 to 15, what's the increase?
The increase is 15-10 = 5 right
So that's technically final value - initial value....
When we need to find increase %, we find the percentage increase on the initial value.. means compared to the initial value what's the increase
In other words we mean to say increase% is:
That if initial value of 10 the increase is of 5
But if the initial value would've been of 100 then what would've been the increase value?
That value is nothing but the value of the increase%
So now let's understand how to derive the formula for increase % through unitary method
I'm hoping you are familiar with unitary method
So when initial value is 10, the increase is 5
When initial value is 1, the increase is 5/10
So if initial value is 100, the increase is (5/10)*100 = 50
And so we say that if the initial value would've been 100, the increase would've been of 50
And that's the formula:
Increase% = ((final -initial)/initial)*100
Now if you check the right hand side
(Final-initial)/initial = final/initial - initial/initial = final/initial - 1(bcoz any value divided by same value gives 1)
So the formula can also be written as
Inc% = ((Final/Initial) - 1)*100
This final/initial is called the Multiplying factor
And (final/initial) -1 is called the increase factor...
So inc% = (final/initial)*100 - 100
That's why we take away the 1 if it's in fraction form OR we take away 100% if it's in % form
Hope this helps....