5 Tips for Faster Mental Division (Part 2)

Have you ever wished you could divide numbers quickly and easily in your head? Believe it or not, you can! Keep on reading to learn my final 2 tips for becoming a mental division maestro.

Jason Marshall, PhD,
August 3, 2013
Episode #164

Mental Division

Quick, what's 54 / 3? How about 324 / 6?

Wait, don't go searching for your calculator or smart phone—believe it or not, you can do these problems and countless others quickly and easily in your head.

In Part 1 of our mental division series, we learned the first 3 tips to becoming a mental division maestro. Today we're going to finish things off with the 2 most powerful mental division tips that are sure to help you kick your calculator dependency forever..

Tips #1-3 Recap

If you haven't yet absorbed, practiced, and perhaps even fallen in love with the first 3 mental division tips that we talked about last time, I highly encourage you to do that before continuing to today's tips. As you'll see, the tricks and tips for today build upon those that we talked about last time, so your efforts will be doubly rewarded.

Before we get to the new stuff, here's a quick recap of what we learned last time::

  • Tip #1—Approximate If You Can: In other words, before pulling out your calculator—or even before you start to use any of the other mental division tricks we've talked about—stop and think about how accurate your answer needs to be. If you only need an approximation, don't waste your time coming up with an exact answer! It's as simple as that.
  • Tip #2—Simplify Before You Start: This is good advice for pretty much anything you do in life, but even more so when it comes to mental divisioin. In this case, it simply means that you should always check to see if you can turn a problem into a simpler problem by dividing out any common factors in the divisor and dividend before starting.
  • Tip #3—Multiply Before Dividing: In our most paradoxical sounding tip, we learned that we can sometimes turn a division problem in which we're dividing by a number like 5 into a simpler problem in which we're dividing by an even power of 10. And the secret is to multiply before we divide. Be sure to check out last week's episode for examples of putting this into practice.

Tip #4: Split the Dividend

We can sometimes turn a division problem, in which we're dividing by a number like 5, into a simpler problem, in which we're dividing by an even power of 10.

Today's 2 tips are really just different incarnations of the same underlying idea. That idea is that we can split-up either the dividend (that's the number you're dividing into) or the divisor (that's the number you're dividing by) to make the problem simpler. Let's start by talking about splitting up the number you're dividing into.

To see how this works, let's think about the problem 54 / 3. Instead of intrepedly setting out to solve this using long division, we're going to split the dividend (that's 54 in this case) into a sum of numbers. In particular, we want to see if we can split 54 into a sum of numbers that are each divisible by 3. Why? Well, it all goes back to the distributive property that we've talked about before which tells us that adding up a bunch of numbers and then dividing the total by some other number is the same as first dividing each number individually by that other number and then adding all of the results together.

In the case of 54 / 3, the trick is to realize that 54 = 24 + 30. And the good news here is that 24 and 30 are both divisible by 3. So according to the distributive property, 54 / 3 = (24 + 30) / 3 = (24 / 3) + (30 / 3). The two problems 24 /3 and 30 / 3 are easy to do in your head, and they immediately tell us that 54 / 3 = 8 + 10 or 18. Pretty cool, right?

Tip #5: Split the Divisor

For the 5th and final trick, instead of splitting up the dividend into a sum, we're going to split the divisor (remember, that's the number you're dividing by) into a product. For example, in the problem 324 / 6, the idea is to realize that 6 = 2 × 3. Which means that 324 / 6 = 324 / (2 × 3). Why is that helpful? Well, I don't know about you, but I have a pretty hard time quickly dividing numbers by 6. But after years of practice, I find it relatively easy to quickly divide by 2 or even 3 (with a bit of extra thinking) in my head. Once you've split up the divisor, you're free to divide the dividend by any of the factors in whatever order is most convenient.

In this case, I'm going to first divide 324 by 3. Why? Well, I know that both 3 and 24 are divisible by 3, so I can almost immediately see that 324 / 3 must be equal to 108. I know that this type of thinking won't be so obvious to a lot of you—I have had a lot of practice with this stuff over the years, after all—but with a bit of work and persistence I promise that it soon will. Since we've figured out that 324 / 3 = 108, all we have to do to solve the initial problem of 324 / 6 is to figure out 108 / 2. It's easy to divide an even number by 2, so we quickly see that 108 / 2 = 54. Which means that 324 / 6 is also equal to 54.

Some of you may have noticed that this trick is very closely related to tip #2 telling us that we should simplify problems by dividing both the dividend and divisor by any common factors. While that's absolutely true, I think it's still useful to think of it as a slightly different trick—a variation on a theme if you will—since it shows that you don't have to first factor both the dividend and divisor (which can sometimes be a pain) to make progress in simplifying your mental division problem-solving life.

Wrap Up

Okay, that's all the math we have time for today.

Be sure to check out my mental math audiobook called The Math Dude’s 5 Tips to Mastering Mental MathAnd for even more math goodness, check out my book The Math Dude’s Quick and Dirty Guide to Algebra.

Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via FacebookTwitter, or email at mathdude@quickanddirtytips.com.

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 courtesy of Shutterstock.

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