Why does $0.75 \times 1.25$ equal $0.93$ and not $1$? If percentage points can equal decimal points, then $1.25$ should equal $125$% when multiplying $1.25$ by any number.
Therefore $125$% of $0.75$ should equal $1$ if $0.75$ is also $75$% of $1$. 
Am I missing something super obvious here or what?!
Please help. Thank you!
 A: The thing you are missing here is assuming that $25\%$ of $0.75$ is $0.25$, when in fact it's $33.\dot3\%$ of $0.75$ that is $0.25$.
$25\%$ of $1$ is $0.25$
A: $$1.25 = \frac 54$$
$$0.75 = \frac 34$$
$$\frac 34 \times \frac 54= \frac {15}{16} = 0.9375$$
Multiplying is not the same as finding the average, which is what you appear to be doing in this case.
A: The mistake you're making is that first taking away some percentage (in this case $25\%$) and then adding the same percentage (again $25\%$) does not give you back the number that you started with. 
For instance, you start with the number $1$. You take away $25\%$, i.e. you obtain $0.75\cdot 1 = 0.75$. Then you add $25\%$, i.e. you obtain $1.25\cdot0.75 = 0.94 \neq 1$. 
The point is that adding $25\%$ here amounts to adding $25\%$ of the reduced value, which is, of course, less than $25\%$ of the original value.
So indeed $0.75\cdot 1.25 \neq 1$, which is correct because $125\%$ of $75\%$ of $1$ is not $1$.
A: Do we agree that, for any number, $x$, $x = \dfrac{75 \% \cdot x}{75 \%}$?  If so, ...
To get what you are wanting to happen to happen, we would need dividing by $75 \%$ to be the same as multiplying by $125 \%$.  But dividing by a fraction is the same as multiplying by its reciprocal:
$$  x = \frac{75 \% \cdot x}{75 \%} = \frac{75 \% \cdot x}{75/100} = (75 \% \cdot x) \cdot \frac{100}{75} = (75 \% \cdot x) \cdot 133.\overline{3}\%  \text{.}  $$
What we have shown is multiplying by $75 \%$ is reversed by multiplying by $133.\overline{3} \%$, not by $125 \%$.
I understand your intuition, that multiplying by $100\% + p\%$ should be reversed by multiplying by $100\% - p\%$, but there are two ways to see that this won't work.  The symbolic way is \begin{align} 
    (100\% + p\%)(100\% - p\%) &= \left(1+\frac{p}{100}\right)\left(1-\frac{p}{100}\right) \\
        &= 1 + \frac{p}{100} - \frac{p}{100} + \left(\frac{p}{100}\right)\left(\frac{-p}{100}\right) & \text{[by distributivity, or by FOILing]}  \\
        &= 1 - \frac{p^2}{10\,000}  \text{,}
\end{align} which isn't quite the same thing as $1$.  The other way to see this is pictorially.  When we multiply a quantity by $75\%$, we break it into four equal parts and discard one of the parts.  When we multiply a quantity by $125\%$ we break it into four equal parts and add a fifth equal part.  If we want to undo multiplying by $75\%$, we start with four equal parts and throw one away, but to get back to the original quantity, we have to add back a copy of one of the three remaining parts (that is, multiply by $100\% + 33.\overline{3}\%$).  If we instead re-partition the three remaining parts into four parts (by erasing the divisions and drawing in new ones), we don't add back quite enough to recover the original quantity.
A: You're right about percentages being equivalent to decimals; they're also equivalent to fractions, and sometimes that's an easier way to think about them.
You can take a percentage and make a fraction out of it by putting it over 1.  I'll do that with $125\%$:
$\frac{125\%}{1} = \frac{1.25}{1}$
But a fraction where one of the numbers has decimal points is kinda hard to work with, so let's multiply this by something that 1) gets rid of the decimal places and 2) is equal to 1, meaning I'll have to multiply it by a fraction that can reduce to 1....meaning the numerator and denominator have to be the same. I'll multiply it by the fraction $\frac{4}{4}$:
$\frac{1.25}{1} * \frac{4}{4} = \frac{5}{4}$
What this means is that "$125\%$ of something" is the same as "$5/4$ths of something." (I can say that they're the same because the only thing I did was to multiply it by something that's equal to $1$, so that didn't change anything except how it looks.)
So let's see what $5/4$ths of $75$ is.  We'll split $75$ up into four pieces, then see what five of those pieces would add up to:
$\frac{75}{4} = 18.75$   <---So that's one fourth of 75.
Now let's see what five of those pieces comes to:
$18.75 * 5 = 93.75$
So there you go: $5/4$ths of $75$ is $93.75$. And since $\frac{5}{4}$ is just the 'fraction' way of writing $125\%$, that means that $125\%$ of $75$ is $93.75$.
A: By this answer, I'm hoping to add a very plain explanation that even children, etc. can understand. (Not targeted at OP, more considering a large audience.)
Let's say you have a glass of juice. You give me a quarter of the juice. You now have 0.75 glass of juice.
Now, you ask me to give you the juice back. But you don't ask me to give you back the same amount of juice. You ask me to give you juice amounting to a quarter of the juice that is currently in your glass. But you only have 0.75 glass in there now! So a quarter of that is only $0.75/4 = 0.1875!$ But you had given me 0.25. So by only asking for that, you cheated yourself out of $0.25 - 0.18715 = 0.0625$ glass of juice.
So the reason is obvious. Once you gave me the juice, you decreased the juice in your glass. Any future multiplication or division of that juice is then obviously affected. So if you wanted to get the same amount back, you would have had to ask me for juice amounting to a third of the juice currently in your glass!
A: Okay. $125\% \cdot 75\%$ is equal to $(100\% + 25\%)\cdot 75\% $
$$ = (100\% \cdot 75\%) + (25\% \cdot 75\%) = 75\% + \frac{75\%}{4} \approx 93\%$$
This is no different than saying $1.25\cdot .75 = 1\cdot (.75) + (.25) \cdot (.75)$
$$\frac 14 \times \frac 34= \frac 3{16}\lt \frac 4{16} = .25$$
A: One way to look at it is to see the decimal numbers as fractions:
$\frac34 \cdot \frac54 = \frac{15}{16} = (.09375)$, not $1$.
It would have to be $\frac34 \cdot \frac43$ to equal $1$.
A: Here is a picture of $0.75 × 1.25$:

Here it is again with the green rectangle in a different place:

A: If you reverse your question it's easier to think about. 
So Think of it like this:
125% of 75% is not 100%
Because:
25% of 75 is not 25 but is 18.75. 
Therefore 
125% of 75 = (75 + 18.75) = 93.75
A: You're correct in your first statement, but another way of thinking of it is to say that multiplying any number by $1.25$ is the same as adding on $\frac14$ to the existing number.  So, $1.25 \times 100 = 100 + 25 = 125$
In the context of $0.75$, a quarter of $0.75$ is $0.1875$, so $125\%$ of $0.75$ is $0.75+0.1875 = 0.9375$.
A fairly simple/basic answer, but I hope it helps!
A: All these answers show that the answer is 0.9375 by a series of manipulations, however I think it is helpful to think why this makes sense.
You are decreasing the first number by 25% to a produce a smaller number. Then this smaller number you are increasing by 25%. Even though you are taking an equal proportion of each number (25%), the initial number was larger so this means that the total size of this proportion is also larger. 
So basically you are taking away a certain amount, then adding a slightly smaller amount.
After you become more experienced with maths these kinds of questions become intuitive and you take it for granted, a very good question though. Good luck with your future mathematical endeavours!
A: You can also see this as the following: $$(1+0.25)(1-0.25)=1^2-0.25^2=0.9375$$
A: When we first learn arithmetic, we look at all the laws of commutation, association and distribution. The list of those principles and laws gets tedious and our intuition summarizes and condenses them as best as it could. 
This condensation process works remarkably great almost always, but the intuition may miss some corner cases.
This may be one such case.
Here, it looks at 0.75 * 1.25 and translates it as
(1 - 0.25) * (1 + 0.25)
and evaluates it to
(1 * 1) + (0.25 - 0.25)
forgetting that the law of distribution requires one more step.
To put it in words, here we  start with a quantity and a fraction x (0.25). We scale the quantity down by multiplying with (1-x).
Then we scale the result back up by multiplying with (1+x). Our intuition then expects to get the original quantity back. 
The mistake here is that we are performing additive inverse by subtracting and adding x, but expecting it to behave like multiplicative inverse which would actually be 1/(1-x).
A: As @wythagoras has already written, the reason is that
$$(1-a)(1+a) = 1 - a^2.$$ 
If $a \neq 1$, then $1 - a^2 \neq 1$. So your idea never works except in the most trivial case ($1\times1 =1$).
Another way to understand this is to ask whether $$\frac{1}{1-a} \overset{?}{=} 1+a$$
In your example, this would mean to ask whether $$\frac{1}{0.75} \overset{?}{=} 1.25.$$
In fact, for small $a$ (and consequently, even smaller $a^2$), this is a fairly good approximation based on Bernoulli's inequality (set $r=-1$ and $x=-a$ in $(1+x)^r \approx 1+rx$ to see that). But in general, it is not true.
To see that, you can calculate the Taylor expansion of $\frac{1}{1-a}$, you find $$\frac{1}{1-a} = 1 + a \color{green}{+ a^2 + a^3 + a^4 + a^5 + ...}$$
The green part is what you were missing.
So when you want to "compensate" $0.75$ (for $a = 0.25$), you could compute $1 + 0.25 \color{green}{+ 0.25^2 + 0.25^3 + 0.25^4 + 0.25^5 + ...} = 1.333 > 1.25$.
It is easy to see that this always work:
$$(1-a)(1 + a + a^2 + a^3 + a^4 + a^5 + ...) \\
= (1 + a + a^2 + a^3 + a^4 + a^5 + ...) \phantom{+(a^6} \\
- \phantom{+(a^6}(a + a^2 + a^3 + a^4 + a^5 + a^6 + ...) \\
= 1.$$
A: So the issue you are having is a misunderstanding of what number is being manipulated - the best way to describe this is to go through the problem step by step.


*

*You start with the number 1

*You take away 25% of 1, giving you 0.75


This is fine, but it's different to the next bit:


*You have 0.75

*You add 25%

*You get 0.9375


Why is this not 1? Because you are taking 25% of 0.75, not one.
25% of 1 is 0.25, but: 
25% of 0.75 is 0.1875
This is why you are getting a different result to what you expect.

Lets look at a modification of the problem so we can see how the problem would have to worded to use your original maths:
You start with 1. You take away 25%. Then you add 25% of your original number.


*

*You start with 1 

*You take away 25%, resulting in 0.75

*You have 0.75

*You add 25% of your original number (25% of 1 is 0.25, so you add
0.25)

*You have 1 (0.75 + 0.25)     



So to be conclude:
With percentages, their value is always relative to another value - they are meaningless on their own. Because of this, they don't hold the same value like an absolute like "12" - percentages change value. You need to be careful of what number it's referencing to - it's usually not the number you started with. Unless it says otherwise, like the second problem I proposed does, it's almost always going to be referencing the last number you calculated.
A: 125% of .75 is not 1.
1.25*.75 can be broken down as:
1 * .75 = .75
then .25 * .75 = .1875
Then add .75 + .1875 = .9375
The key is in the line .25 * .75 = .1875
To bring percentages back into it, 25% of .75 equals 1/4th of .75. What you're thinking is that .25 of .75 is 1/4th of .75. 
so what is 1/4th of .75? it's .1875.
25% of .75 is also 1/4 of .75, which is .1875.
