If you’re anything like me, you don’t exactly love doing long division. Which is exactly why I avoid it as much as I can. Of course, one way to avoid doing division the old fashioned way with paper and pen is by using a calculator. Most of the time, that’s exactly what I do.
But the truth is that sometimes calculators—or phones with calculators—are inconvenient. And sometimes you need to do division right there on the spot in your head. How can you do it? Keep on reading to learn 5 simple things that you can do to take your mental division skills to the next level.
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Tip #1: Approximate If You Can
The first thing you can do to speed up a lot of the mental division problems you’ll encounter is to stop and think about just how accurate you need the answer to be. Sometimes you need an exact answer, or perhaps an answer that’s accurate to two decimal places, or three decimal places, or something else specific like that. But a lot of the time you really just need a ballpark estimate.
If you only need an approximate answer, don’t waste time figuring out the exact answer…make a quick and dirty estimate instead.
In those cases where you only need an approximate answer, don’t waste your time by figuring out the exact answer. Instead, make a quick and dirty estimate. How? Well, let’s say you work in a coffee shop and you want to figure out the average amount spent by your customers. So far you’ve collected $164 from 26 people. What’s the average—or more technically the mean—bill? Well, $164 is pretty close to $150, and 26 people is pretty close to 25 people. So instead of calculating $164 / 26 people, let’s start by calculating something that’s close to that: $150 / 25 people. That’s a much easier problem to solve! It says that the answer is roughly $6 / person.
No, this technique doesn’t give exact values, but if you estimate wisely it’ll give you answers that are pretty close…and with a fraction of the work.
Tip #2: Simplify Before You Start
The rest of today’s tips are all things that you can do when an approximate answer just isn’t good enough. When faced with such problems, the first thing you should do is…
…check if you can simplify the problem before you start doing any division. In particular, if the number you’re dividing into (aka, the dividend) and the number you’re dividing by (aka, the divisor) are both even, you should start by dividing both by 2. For example, in the problem 164 / 26, start by dividing both the top and bottom by 2 to obtain the simplified (but otherwise identical) problem 82 / 13. Why is that helpful? Simply because it’s almost always easier to work with smaller numbers.
But that’s not the only way a division problem can be simplified. In a problem like 93 / 27, the dividend and divisor are not both even numbers—so we can’t simplify the problem by dividing both by 2. But both are divisible by 3 (in other words, 3 is a factor of both 93 and 27). Which means that in this case we can divide both the dividend and divisor by 3 to obtain the simplified problem 31 / 9.
The quick and dirty tip here is to always look for factors common to both the dividend and the divisor, and then to divide both by each common factor before you do anything else.
Tip #3: Multiply Before Dividing
There are times in life when doing the opposite of what you should be doing is bad. But there are other times when it’s actually very, very good. This is one of those times because when it comes to mental division, it turns out that it’s often helpful to multiply instead.
Like many mental math tricks before it, this one relies upon the fact that dividing by 10 (or 100, or any other power of 10) is easy—just move the decimal point one position to the left for each power of 10. And since it’s so easy to divide by powers of 10, we should strive to do it as often as possible—even when we’re supposed to be doing something like dividing a number by 5. But how can we possibly do that? We use the fact that 5 = 10 / 2.
For example, let’s imagine you’re back working in your coffee shop and you’ve decided to calculate the average amount of money you make each day of a 5-day work week. In other words, you need to divide the total amount of money you bring in over those 5-days—let’s say it’s $1,677—and divide it by 5. But instead of calculating $1,677 / 5, let’s use the fact that 5 = 10 / 2 to turn this problem into $1,677 / (10 / 2). Using what we’ve learned about how to divide fractions (remember invert and multiply?), we see that this is the same as $1,677 × 2 / 10. In other words, we’ve managed to turn a hard problem of mentally dividing by 5 into two simple problems. First, multiply $1,677 by 2 to get $3,354. And second, divide the result by 10 to find that the average income for each day of the week is $3,354 / 10 = $335.40.
Does this trick only work for division by 5? No! You can use the same idea to turn division by 20 into multiplication by 5 followed by division by 100, or division by 25 into multiplication by 4 followed by division by 100, and lots and lots of other things, too. And when combined with today’s first tip about making approximations, this trick becomes even more powerful. Just remember to stop and think before you start working on a problem, and you’ll often see that there’s a much easier way to the solution.
Wrap Up
Okay, that’s all the math we have time for today. Of course, those are only 3 of my top 5 tips for faster mental division. So be sure to check back next time to learn the last—and most powerful—pair of tips!
Be sure to check out my mental math audiobook called The Math Dude’s 5 Tips to Mastering Mental Math. And for even more math goodness, check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
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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!
Mental division image from Shutterstock.