Amazing Multiplication Trick
Tip #1: Multiplying by Powers of 5
There are times in life when you just get lucky. It turns out that one of those lucky little moments occurs each and every time you need to multiply one number by another number that happens to be a power of 5. For example, let’s say you need to find 36 x 5 (which, of course, fits the bill since 5 is the first power of 5). The trick is to recognize the fact that 5 = 10 / 2. Why is that helpful? Because it means that we can find 36 x 5 by instead finding 36 x 10 (which is easy) and then dividing the result by 2. In this case, 36 x 10 = 360, and 360 / 2 = 180. Impressively speedy, right?
But we’re not done! What if we instead need to solve the problem 36 x 25? Well, this trick is all about multiplying by powers of 5…and 25 = 5^2 is certainly that. So how does it work in this case? The trick here is to recognize that 25 = 100 / 4. And in general, the trick with powers of 5 is to recognize that they are always some multiple of 10 divided by an integer. This tells us that 36 x 25 = 36 x 100 / 4. Since we can quickly figure out that 36 x 100 = 3,600, it’s easy to find that 36 x 25 = 3,600 / 4 = 900.
Tip #2: Squaring Numbers Ending in 5
Our fun with 5s doesn’t end there. We talked about how to square numbers in your head before, but it turns out that things get a whole lot easier when squaring a two-digit number that ends in 5. Here’s the trick: Any time you square a two-digit number that ends in 5, the last digits of the answer will be 25 and the digits before that are given by multiplying the first digit of the number by the number that’s one greater.
For example, this trick says that the last two digits of 45^2 must be 25, and the digits before that are given by 4 x 5 = 20. So 45^2 = 2,025. How about 75^2? Well, once again we know that the last two digits will be 25 (since they always are for this kind of problem), and the previous digits are given by 7 x 8 (that’s the first digit times the number that’s one greater). So the answer is 75^2 = 5,625. Fast and easy!
Tip #3: Easily Multiplying Lots of 9s
The third trick for today has to do with multiplying any number by 9, 99, 999, or any other number that’s 1 less than a power of 10. What makes all of these wild 9 numbers special? In a problem like 44 x 9, the trick is to recognize that 44 x 9 = 44 x (10 – 1). The distributive property of multiplication tells us that this is the same as 44 x 10 – 44. And since it’s easy to multiply by a power of 10, looking at the problem this way makes it much easier to solve. In particular, it tells us that 44 x 9 = 44 x 10 – 44 = 440 – 44 = 440 – 40 – 4 = 396 (sharp-eyed math fans may notice a trick there related to the mental subtraction tips from before).
If we’re instead trying to solve 44 x 99, the trick is to recognize that this is the same as 44 x (100 – 1) = (44 x 100) – 44. In other words, any time you’re multiplying by one of these numbers that are all 9s, the trick is to know that you can simply multiply the other number by the next higher power of 10 and then subtract the original number. Give it a try and you’ll see just how much faster this is.
If you’re trying to solve 44 x 99, the trick is to recognize that this is the same as 44 x (100 – 1) = (44 x 100) – 44.
Tip #4: Multiplying by Powers of 2
You can use today’s fourth tip any time you’re multiplying one number by another number that’s a power of 2. Which means that any time you’re multiplying some number by 2, 4, 8, 16, 32, 64, and so on, this is your ticket to mental math bliss. Instead of going through the usual multiplication process, in this case all you have to do is double the number you’re multiplying for each power of 2 in the other number.
For example, the problem 12 x 8 is the same thing as 12 x 2^3 or 12 x 2 x 2 x 2. Which means that we can quickly find the answer by continually doubling 12 three times. So the first doubling of 12 gives 24, the second doubling takes us to 48, and the third doubling gives 96. So 12 x 8 = 96.
Tip #5: Double and Halve to Multiply Fast
The previous trick is really just a special case of today’s fifth and final (and I think coolest) trick that you can use whenever one of the numbers you’re multiplying is even. Let’s say you’re multiplying 47 x 24. Since 24 is an even number, let’s use the idea of doubling and halving to solve this problem quickly.
What do I mean by doubling and halving? Well, the trick is to continually double one number while halving the other. In this case, this means that we turn the problem 47 x 24 into the problem 94 x 12 by simultaneously doubling 47 and halving 24. We can then do the same thing and turn the problem into 188 x 6, and again to get 376 x 3. At this point, we can’t double and halve any further, so we just have to do the remaining—much easier!—multiplication problem to find that 47 x 24 = 376 x 3 = 1,128.
You’ll definitely need to practice these techniques to get comfortable (and fast) using them—so I highly encourage you to make up some multiplication problems to work through. It will take some time and energy, but your effort will certainly be rewarded!
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 Math. And 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 Facebook, Twitter, or email at firstname.lastname@example.org.
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 math image from Shutterstock.