Percentages Are Reversible
What do we mean by reversible?
We're asked what is 20% of 50.
We could reverse the numbers to make the problem a bit easier.
That is, 50% of 20 is the same as 20% of 50.
Why are they reversible? Seems like magic, but it's far from it.
It's actually simple. Percentage means per hundred.
Whenever we see a percentage, we can replace it with (1/100) or .01.
So, when we are asked what is 20% of 50 we are really doing 2 multiplications.
20 * 50 * .01 = 10
Key Takeaways
Multiplication is Commutative.
Commutative means that order doesn't matter.
a * b = b * a
An example of an operation that is not commutative is subtraction.
a - b != b - a
Therefore, since taking a percentage involves just two multiplications and a percentage is .01 the order of operations doesn't matter.
(20 * .01) * 50 = 20 * (50 * .01) = 10
Conclusion
Reversing a percentage may not always make the problem easier, for example
7% of 13 and 13% of 7 are about equally difficult to solve without a calculator.
However, in some problems it can make life simpler, but what is more important is for us to understand what a percentage really is and why we can reverse it.
Often times in mathematics, and in life, if we really understand a problem, we can rearrange it to our advantage.
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