Holy cow, I wrote a book!
The Windows calculator percent sign works the same way as those
cheap pocket calculators (which are often called four-function calculators
even though they have around six function nowadays).
What you first have to understand is that the percent key on those
pocket calculators was
not designed for mathematicians and engineers.
It was designed for your everyday person doing some simple calculations.
Therefore, the behavior of the key to you, an engineer,
seems bizarrely counter-intuitive and even buggy.
But to an everyday person, it makes perfect sense.
Or at least that's the theory.
Let's look at it from the point of view of that everyday person.
Suppose you want to compute how much a $72 sweater will cost
after including 5% tax.¹
Pull out your handy pocket calculator² (or fire up Calc if you don't
have a pocket calculator) and type
72 + 5% =
The result is 75.6, or $75.60,
which is the correct answer, because 5% of 72 is 3.6.
Add that to 72 and you get 75.6.
Similarly, suppose that sweater was on sale at 20% off.
What is the sale price?
72 − 20% =
The result is 57.6 or $57.60.
This is the correct answer,
because 20% of 72 is 14.4.
Subtract that from 72 and you get 57.6.
You can chain these percentage operations, too.
For example, how much will you have to pay for that 20%-off sweater
after adding 5% tax?
72 − 20% + 5% =
The result is 60.48.
A mathematician or engineer would have calculated the same result via
the equivalent computation:
72 × 0.80 × 1.05 =
Okay, now that we see how the calculator product designer intended the
percent key to be used,
let's look at what the calculator engineer
it has to do in order to match the specification.
When the user enters A + B % =,
the result should be
A × (1 + B/100) or
A + (A × B/100) after you distribute the multiplication
over the addition.
Similarly, when the user enters A − B % =,
the result should be
A × (1 − B/100) or
A − (A × B/100).
Aha, the calculator engineer says,
we can achieve this result by defining the percent key as follows:
When the user enters a value, an operator, a second value,
and then the percent key,
the first two values are multiplied and the product divided by 100,
and that result replaces the second value in the ongoing computation.
Let's walk through that algorithm with our first example.
If you watch the display as you go through this exercise,
you will even see the number 3.6 appear in the display
once you press the % key.
The percentage is calculated and replaces the original value
in the ongoing computation.
This algorithm also works for the chained percentages.
This even works for multiplication and division, but there is
much less call for multiplying or dividing a number
by a percentage of itself.
500 × 5 % =
The result of this is 12,500 because you are multiplying 500
by 5% of 500 (which is 25).
The result of 500 × 25 is 12,500.
You aren't computing five percent of 500.
You're multiplying 500 by 5% of 500.
(It appears that the authors of
this Knowledge Base article didn't consult
with the calculator engineer before writing up their analysis.
The percent key is behaving as designed.
The problem is that the percent key is not designed for engineers.)
What if you want to compute 5% of 500?
Just pick a dummy operation and view the result when you press
the percent key.
500 + 5 %
When you hit the percent key, the answer appears: 25.
You could've used the minus key, multiplication key, or division key
instead of the addition key.
It doesn't matter since all you care about is the percentage,
not the combined operation.
Once you hit the % key, you get your answer,
and then you can hit Clear to start a new calculation.
¹In the United States, quoted prices typically do not include
²In my limited experiments, it appears that no two manufacturers of
pocket calculators handle the percent key in exactly the same way.
Casio appears to handle it in a manner closest to the engineering way.
TI is closer to the layman algorithm.
And when you get into cases like 1 ÷ 2 %,
calculators start wandering all over the map.
Should the answer be 50, since 1/2 is equal to 50%?
Or should it be 0.005 since that is the numeric value of 0.5%?
Should that answer appear immediately or should it wait for you to hit the
I don't know what the intuitive result should be either.
1 ÷ 2 %