4 More FAQs About Percentages
How do you quickly calculate 25% of a number? Or 33% of a number? Or 50%? And how can you quickly calculate percentage increases? Keep on reading to learn the answers to these frequently asked questions about percentages.
We here at the Math Dude ranch get numerous questions every week from math fans around the world. By far the most common questions we receive have to do with calculating percentages. In particular, how to quickly calculate percentages in your head. You know, things like: What’s 25% of $14,000? Or what’s the final price after a 33% discount on a $25 item? Or what’s the percentage increase from 30 to 40?
It’s not hugely surprising that this is such a popular line of questions since people in lots of different industries love to express changes in terms of percentages. So today we’re going to take a look at four of the most frequently asked questions about percentages.
Holiday Puzzle Solution
But before we dive into percentages, I want to fill you in on the solution to the puzzle I posed last time. Actually, it’s the puzzle tweeted by psychologist, magician, and guest of the show Richard Wiseman. In case you’ve forgotten, here’s how it works. First, grab a calculator. Then do the following:
- Type your house number (i.e., your address) into a calculator.
- Now double it.
- Next add 5 to the result.
- Then multiply this answer by 50.
- Now add your age.
- And then add 365 to the result.
- Finally, subtract 615 from the whole thing.
What do you get? If you did it right, you should see your house number and age (so long as you’re under 100 years old). Why? It’s actually fairly simple to understand with a bit of algebraic thinking. To begin, let’s call your house number “A” and your age “B”. If you follow the steps in Richard’s tweet, you’ll see that the whole sequence of actions is equivalent to the algebraic expression:
(((((2 x A) + 5) x 50) + B) + 365) – 615
Which is quite a mess! How does it help us make sense of the trick? Well, if we simplify the expression a bit, we see that we can combine and arrange the terms to turn it into the equivalent expression:
((2A + 5) x 50) + B – 250
Admittedly, this isn’t much better, but if we simplify this even more we find that we can multiply and then combine terms to arrive at a much simpler equivalent expression:
100A + B
And now we’re getting somewhere. Because this expression tells us that all you’re really doing is multiplying your address by 100 (which has the effect of padding the end of it with a pair of zeros) and then adding your age (which has the effect of sticking it on the end). Once you know this, you can see that all of the complicated actions were simply a distraction to keep you from noticing the simple thing happening when you weren’t looking. In other words, it’s a magic trick.
Percentage Tip 1: Calculating 10%, 20%, 30%, …
Math fan Amanda recently wrote to ask about calculating 10% of a dollar amount. She wrote “Every time I do this, I have to Google for a reminder and never end up working it out the same way twice. So what’s the easy way?”
Well Amanda, the good news is that all you need to remember to make sense of percentages is that the word “per-cent” means “per 100” (“cent” also shows up in century, centipede, and the 1 cent value of the United States penny). So 1 percent of some value is the same as the fraction 1/100 of that value. And 10 percent of some value is the same as 10/100 = 1/10 of that value.
As such, calculating 10% of a value is really, really easy to do in your head. All you have to do is divide the value by 10. So 10% of $90 is just $90/10 = $9. And it’s not much tougher to calculate 20% or 30% of a number. Simply calculate 10% of the number and then multiply the result by 2 or 3. So 20% of $90 is 2 x $9 = $18, and 30% of $90 is 3 x $9 = $27. You can use the same logic to find 40%, 50%, 60%, or any other similar percentage.
Percentage Tip 2: Calculating 25%, 33%, 50%, 66%, and 75%
Several math fans have written asking how to calculate other percentages such as 25%, 33%, 50%, 66%, and 75%. The good news is these are also easy to calculate in your head.
To begin with, since 25% is the same as 25/100=1/4, you can calculate 25% of any number by dividing it by 4. So 25% of $60 is just $60/4=$15. Similarly, since 33% is the same as 33/100 or approximately 1/3, you can calculate 33% of a number by dividing it by 3. And since 50% is the same as 50/100=1/2, you can calculate 50% of a number by dividing it by 2.
The percentages 66% and 75% are almost as easy. Since 66% is the same as 66/100 or roughly 2/3, you can calculate 66% of some number by dividing it by 3 and then multiplying the result by 2 (or the other way around if it’s easier). And since 75% is the same as the fraction 75/100=3/4, you can calculate 75% of a number by dividing it by 4 and then multiplying the result by 3.
Percentage Tip 3: Calculating Fractional Percentages
Math fan Amirah wrote: “I have always had trouble calculating percentages quickly in my head; I can do it slowly on paper but it’s not the most efficient method. Do you have any tips for calculating decimal percentages quickly? For example, what’s 0.001% of $1,000,000?”
Fortunately for Amirah, calculating percentages that are a fraction of 1% isn’t all that difficult once you understand how to calculate 10% of a number. This is most easily seen by working out Amirah’s example of finding 0.001% of $1,000,000.
The trick is that for each factor of 10 smaller percentage, we divide the number by yet another factor of 10. Remember that to calculate 10% of a number we divide by 10. And to calculate 1% of a number we divide by 100. To calculate smaller percentages, we just continue this trend. So to calculate 0.1% we divide by 1,000; to find 0.01% we divide by 10,000; and to find 0.001% we divide by 100,000. So 0.001% of $1,000,000 is $1,000,000/100,000=$10.
Percentage Tip 4: Calculating Percentage Increase
Finally for today, math fan Matthew recently wrote: “In the course of my work, I read sentences such as ‘Some company delivered another strong quarter with sales of $9.4 billion, or an increase of 6%.’ What’s a quick and dirty way to ballpark what the original number was?”
The trick that helps me here is to translate this into a very simple equation in my head. The equation says that 1.06 (which is the same as 106%) times some number is equal to $9.4 billion. The question is what’s the number? If you think about it, you’ll see that all we have to do to find out is divide $9.4 billion by 1.06. In general, you can always solve a problem like this to find the initial number by dividing the final number by the percentage increase or decrease (represented as a decimal number). Once you know the trick and where it comes from, you should never need to write the equation down again.
Wrap Up
Okay, that’s all the math we have time for today.
For more fun with math, please check out my book,The Math Dude’s Quick and Dirty Guide to Algebra. Also, remember to become a fan of The Math Dude on Facebook and to follow me on Twitter.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
FAQ image from Shutterstock.
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.
