Concepts I touch on that you should be familiar with:

• Finding the percent of a number
• Knowing 50% is half, 25% is a quarter, 75% is ¾, etc. 
• Slight algebraic manipulation 

In school, you’ve probably learned how to find 10% of 25 or 50% of 4 or something along those lines. While you may be able to do 50% of 4 in your head, as 50% means half and half of 4 is 2, other numbers may require more calculations. For example, what is 7% of 50? Well, you can change 7% into 0.07 and multiply that by 50, but no one wants to calculate that, especially without a calculator. Well for those who know this rule, 7% of 50 becomes very straightforward.

Let’s take the first example of 10% of 25. 10% is 0.1, and 0.1 times 25 is 2.5. 

Now let’s look at 25% of 10. 25% is a quarter, and therefore 10 divided into 4 quarters is 2.5.

That’s no coincidence. You can reverse the percentage when you do these problems! That’s why 7% of 50 is much easier than it looks. 50% of 7 is just half of 7, which is 3.5! 

Now, why does this work? What’s the algebra behind this? 

Let’s first convert the percentages to decimals:

7% is 0.07, or 7/100. 50% is 0.5 or 50/100. 

We can write 7% of 50 as 7/100 times 50.

We can write 50% of 7 as 50/100 times 7.

If you then split each fraction into the numerator times 1/100, you’ll find each of the equations to be the same, due to the commutative property of multiplication (the order of the numbers being multiplied doesn’t matter). Isn’t that awesome? 

Now you can impress your friends by solving seemingly impossible percentage problems in your head. 4% of 75? Easily 3. 12% of 33 and a third? Well, that’s 4 of course!