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?!

• You seem to be asking why $125\%$ of $75\%$ is not $100\%$. Question is why do you think that? – JDF Sep 20 '16 at 13:36
• If I take half of your money and give you back 50% of the money you still have, how much money do you have? Even worse: If I take all of your money and give you back the same amount of money you still have, you will have nothing at all. – ctst Sep 20 '16 at 13:37
• Your math is correct, you're just making a mathematical fallacy: adding X% and removing X% are not inverse operations. This is a widespread misunderstanding - I see this mistake made eg. in the news all the time. – BlueRaja - Danny Pflughoeft Sep 20 '16 at 15:25
• In my opinion, percentages are so often ambiguous, abused, and simply misunderstood that they are basically worthless for precise communication or for calculation. – Excluded and Offended Sep 20 '16 at 16:17
• To connect this with a well-known school algebra identity, note that $125$% of $0.75$ can be written as $(1 + \frac{1}{4})(1 - \frac{1}{4}),$ which we now immediately recognize as being $1 - \frac{1}{16}$ (in particular, less than $1)$ from the difference of squares factorization formula $(a+b)(a-b) = a^2 - b^{2}.$ (moments later) Oops, I see @wythagoras (answer below) has beat me to this! – Dave L. Renfro Sep 20 '16 at 16:19

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 $1$.

• +1 for pointing out that the second 25% is 25% of the reduced value. This is a key point that most other answers cover only by implication. – phoog Sep 20 '16 at 18:47
• You could point out that the right way to "undo" the multiplication of a percentage, the reverse of the operation, is not to multiply by 125%, but to divide by the percentage. If 1 * 75% = 0.75, then to undo that operation, it is 0.75 / 75% = 1. I would write this in my own answer, but there are too many answers already. (Also notice that this is the same as multiplying by the inverse of 75%, and that that value is not 125%, but 4/3 ~= 133.333...%) – Dave Cousineau Sep 20 '16 at 21:29
• God I'm so vacant. Suits me right for doing homework at 1am! Thank you Sjors, makes perfect sense! – Joshua Aimless Blameless Sep 21 '16 at 2:40
• I think you understood what the questioner was asking - why you don't get the average of two numbers when you multiply them together. Perhaps a simple example would help illustrate the fallacy. Why doesn't (4-1)*(4+1)=4? Why doesn't (1-1)*(1+1) =1 i.e why doesn't 0*2=1? – Readin Sep 21 '16 at 3:54

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!

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!

• Well stated. A clear explanation of what the OP was overlooking in his naive analysis. – Paul Sinclair Sep 20 '16 at 17:21
• It also works if you change the order and add 25%, then subtract 25%: You add 25%, you get a larger number. Then you subtract 25% of a larger number, so you end up with less than you had originally. – gnasher729 Sep 20 '16 at 21:19

You can also see this as the following: $$(1+0.25)(1-0.25)=1^2-0.25^2=0.9375$$

• I remember looking at the multiplication table in elementary school and noticing that taking a line perpendicular to the main diagonal (squares) deviated from that diagonal by successively-increasing squares. Example: 8*8=64 but 9*7=63 (1 less) and 10*6=60 (4 less) 11*5=55 (9 less) etc. I pointed this out to the teacher because it looked cool. She smiled and agreed. It was years later that I learned the algebra behind why this must be true. – Monty Harder Sep 20 '16 at 19:18
• this is how I think of it! but I admit I've confused many early learners by using this approach – MichaelChirico Sep 21 '16 at 4:15

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 $0.25$

$$1.25 = \frac 54$$

$$0.75 = \frac 34$$

$$\frac 34 \times \frac 54= \frac {15}{16} = 0.9375$$

Multiplying is not the same as finding the average, which is what you appear to be doing in this case.

• Certainly one of the simplest and most obvious explanations. – Adrian Larson Sep 20 '16 at 21:00

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.

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 $93.75$.

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 $1.25\cdot .75 = 1\cdot (.75) + (.25) \cdot (.75)$

$$\frac 14 \times \frac 34= \frac 3{16}\lt \frac 4{16} = .25$$

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 milk. You give me a quarter of the milk. You now have 0.75 glass of milk.

Now, you ask me to give you the milk back. But you don't ask me to give you back the same amount of milk. You ask me to give you milk amounting to a quarter of the milk 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 milk.

So the reason is obvious. Once you gave me the milk, you decreased the milk in your glass. Any future multiplication or division of that milk is then obviously affected. So if you wanted to get the same amount back, you would have had to ask me for milk amounting to a third of the milk currently in your glass!

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 $1$.

Here is a picture of $0.75 × 1.25$:

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

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

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: $$(1-a)(1 + a + a^2 + a^3 + a^4 + a^5 + ...) \\ = (1 + a + a^2 + a^3 + a^4 + a^5 + ...) \phantom{+(a^6} \\ - \phantom{+(a^6}(a + a^2 + a^3 + a^4 + a^5 + a^6 + ...) \\ = 1.$$

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.

2. You take away 25% of 1, giving you 0.75

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

1. You have 0.75
3. 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:

2. You take away 25%, resulting in 0.75
3. You have 0.75
4. You add 25% of your original number (25% of 1 is 0.25, so you add 0.25)
5. 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.

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).

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.

• You may consider using MathJax to improve your formatting. Up to now it's really hard to read your answer. – Cave Johnson Sep 21 '16 at 3:46

protected by Alex M.Sep 21 '16 at 16:51

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