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Quick Tricks for Multiplication Why multiply? A computer can multiply thousands of numbers in milliseconds. A human is

lucky to multiply two numbers in less than a minute. So we tend to have computers do our math. Banks use computers. Governments use computers. Brilliant mathematicians use computers.

Yet your math teacher still makes you haul out pen and paper and multiply. She even suggests you do it in your head. Should this be legal?

Well, yes. You have to know a little math even to use a calculator. Besides, daily life tosses plenty of math problems your way. Do you really want to haul out Trusty Buttons every time you go shopping?

Knowing how to do basic math is like knowing how to walk. You could crawl to the car every time you wanted to go more than fifty feet, but you'd get flabby. For a short distance, a walk is healthy, interesting, and even fun.

Still, normal multiplication can get boring. Here's the secret: shortcuts. You might think of numbers as a dreary line from 0 to forever. Numbers do go on forever, but you can also think of them as cycles. Ten ones make 10. Ten tens make 100. Ten hundreds make 1000.

If numbers were just a straight highway, there'd be no shortcuts. But they're more like a winding road. If you know your way around, you can cut across the grass and save lots of time.

Multiply by 10: Just add 0 The easiest number to multiply by is 10. Just “add 0.”

3 x 10 = 30 14 x 10 = 140 13,260 x 10 = 132,600 With decimals, you move the decimal point once to the right.

3.41 x 10 = 34 .1 14.0 x 10 = 140 0.01326 x 10 = 0.1326 This “trick” is really just using our number system. 3 means “3 ones.” Move 3

once to the left and you get 30. You had 3 ones, now you have 3 tens. See how our numbers cycle in tens? Whenever you move the digits once to the left, that's the same as multiplying by 10.

And that's the quick way to multiply by 10. Move each digit once to the left. (You can also think of this as moving the decimal point to the right if that's easier.) If that leaves an empty space, fill in a 0.

How about dividing by 10? Just move each digit once to the right. 140 / 10 = 14 13.41 / 10 = 1.341

Easy.



Exercise A: 1. Give two reasons to get good at doing math in your head.

2. The author compares doing math on a computer to using a car. If you use a car

for “long distances,” what kind of math problem might you use a computer for? Give two examples.

3. Give two examples of situations where it might be more convenient to do math in

your head than with a calculator.

4. How do you quickly multiply by ten? Why does this work?

5. Solve these problems without using a calculator.

a. 4 x 10 b. 0.004315 x 10 c. 4.00 x 10 d. 139,200 x 10

6. How would you quickly multiply by 100?

7. How would you quickly multiply by 10,000? 8. Solve the following problems without using a calculator.

a. 83.1 x 1,000 b. 0.0357 x 100 c. 9.10 / 10,000 d. 204 / 10 e. 453.09 / 100 f. 78 / 100



Multiply by 9: It All Adds Up to 9 Have you ever heard of the Amazing Facts of Nine? Although they sound like an

unpromising title for an afterschool miniseries, the Amazing Facts of Nine are both factual and amazing. Let's take a look.

2 x 9 = 18 1 + 8 = 9 3 x 9 = 27 2 + 7 = 9 4 x 9 = 36 3 + 6 = 9 5 x 9 = 45 4 + 5 = 9 6 x 9 = 54 5 + 4 = 9 7 x 9 = 63 6 + 3 = 9 8 x 9 = 72 7 + 2 = 9 9 x 9 = 81 8 + 1 = 9

Sensing a pattern? When we multiply a single-digit number times 9: - The tens digit is one less than our original number. - The tens digit plus the ones digit equals nine! This makes it ridiculously easy to multiply any single digit times 9. Suppose your

quirky uncle barks, “Multiply 5 times 9!” and, in the heat of the moment, your lovingly memorized multiplication table deserts you. Never fear. First, the tens digit is one less than 5. (5 is the multiplicand.)

5 - 1 = 4 subtract 1 from multiplicand to get tens digit Second, you know that this tens digit (4) plus the ones digit will equal 9. Find the

ones digit by subtracting the tens digit from nine. 9 - 4 = 5 subtract tens digit from 9 to get ones digit

Eureka! The answer is 45. (And you can double check.) 4 + 5 = 9 double check, digits add up to 9

Why does this work? Well, 9 is the highest digit in our number system. Look at the 9 facts above. When you add 9, you subtract 1 from the ones digit and add 1 to the tens digit. So the digits still add up to that same number 9.

Of course, you'll be sure to engrave the multiplication table in your heart anyway. But this is a handy trick. Remember, it only works for single digits. Once you get past single digits, it changes.

53 x 9 = 477 4 + 7 + 7 = 18 1 + 8 = 9 4689 x 9 = 42201 4 + 2 + 2 + 0 + 1 = 9

If you keep adding the digits, you still get back to 9. This won't help you solve a problem, but you can use it to check your work. Plus: it's weird and cool.



Exercise B: 1. Explain the quick way to multiply a single digit times 9. 2. Solve the following problems without a calculator.

a. 4 x 9 b. 9 x 9 c. 9 x 8 d. 5 x 9 e. 7 x 9 f. 9 x 2 g. 3 x 9 h. 6 x 9

3. Do you think it's true that you can multiply any number times 9, and the digits will eventually add up to 9? Test this theory with the following numbers. (You may use a calculator.)

a. 36 b. 252 c. 5,817 d. 157,931 e. 91,572.425

4. Nothing in math happens by accident. Our theory is that this trick works because 9 is the highest digit in our number system. We can test it by trying another number system. Suppose we counted like this:

1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 ...

This is called octal. Here, “10” means eight. The “tens” digit is the “eights” digit. 10 means 1 eight, no ones. 11 means 1 eight and 1 one—nine. 12 means ten, and so on. In this system, the highest digit is 7. So multiplying by 7 in octal should be like multiplying by 9 in our system.

Test this theory by multiplying 7 times 2, 5, and 6 in octal. (You can find the answer in our system, then change it into octal.) Show your work on the back of this page. Does this theory seem true or false?

Do you have a calculator that multiplies in octal? Try multiplying 7 by even bigger numbers. Do these answers have digits that eventually add up to 7?



Multiply by 5: It's All 5s and 0s Lots of folks know it's easy to multiply by 10. Just shift the digits once to the left.

If necessary, add 0. 2.1 x 10 = 21 2 x 10 = 20

But what about 5? Let's look at a few facts: 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20 5 x 5 = 25 6 x 5 = 30 7 x 5 = 35

See a pattern? Whenever we multiply by 5, if the multiplicand is even, the answer ends in 0. If it's odd, the answer ends in 5. It doesn't matter how big the number is.

1,652 x 5 = 8260 1,653 x 5 = 8,265 This is a handy way to check your work. If you multiply 3,73 by 5 and get 1,677,

you know right away that all is not well in Mathland. But is there a trick to multiply by 5 quickly in the first place?

Yes. Remember how easy it is to multiply by 10? Well, 10 = 2 x 5. So when you multiply by 10, you can think of it as multiplying by 5 and then by 2.

4 x 10 = 40 4 x 5 = 20 and 20 x 2 = 40 What if you meant to multiply by 5, but you accidentally went too far and

multiplied by 10? 4 x 5 = 20 20 x 2 = 40 (oops, too far!)

You could just undo the second step. 40 / 2 = 20 (same as 4 x 5 = 20)

You multiplied the number by 10, then divided it by 2. That was the same as multiplying it by 5. Hmm. You can do that backwards.

To multiply by 5, divide by 2, then multiply by 10. 6 x 5 = 30 6 / 2 = 3 and 3 x 10 = 30 7 x 5 = 35 7 / 2 = 3.5 and 3.5 x 10 = 35 167 x 5 = 835 166 / 2 = 83.5 and 83.5 x 10 = 835

Of course, if a number's too big, it might be almost as hard to divide by 2 as to multiply by 5. But whenever it's easy for you to divide by 2, it's just as easy to multiply by 5. Remember, an odd number divided by 2 will always end in .5, so you can subtract 1, divide that even number, then add 0.5.

17 / 2 = ? 17 – 1 = 16 and 16 / 2 = 8, so 17 / 2 = 8.5



Exercise C: 1. What's a quick way to multiply a number by 5? 2. Explain how this trick works. 3. Solve these problems without a calculator.

a. 4 x 5 b. 12 x 5 c. 19 x 5 d. 15 x 5 e. 7 x 5 f. 88 x 5 g. 230 x 5 h. 21 x 5

4. Don't solve these problems. Instead, check whether each answer could be right or not. Explain your answer.

Example: 4 x 5 = 25 No. An even number x 5 should equal a number that ends in 0. a. 11 x 5 = 50 b. 3 x 5 = 18 c. 14 x 5 = 70 d. 217 x 5 = 1080



Multiply by 3: It All Adds Up to 3 (kind of) Remember the Amazing Facts of Nine? When you multiply by 9, the digits of the

answer eventually add up to 9. 8 x 9 = 72 7 + 2 = 9 53 x 9 = 477 4 + 7 + 7 = 18 1 + 8 = 9

So how about that number 9? It's 3 times 3, isn't it? Since 9 has such nifty qualities, you'd think that 3 might be a bit unusual, too.

2 x 3 = 6 These single-digit answers seem normal. 3 x 3 = 9 But when we get to 2 digits ... 4 x 3 = 12 1 + 2 = 3 5 x 3 = 15 1 + 5 = 6 6 x 3 = 18 1 + 8 = 9 7 x 3 = 21 2 + 1 = 3

Whoa! It looks like whenever we multiply a number by 3, the digits of the answer add up to 3, 6, or 9. Let's check a couple really high numbers.

524 x 3 = 1572 1 + 5 + 7 + 2 = 15 and 1 + 5 = 6 91,317 x 3 = 273,951 2 + 7 + 3 + 9 + 5 + 1 = 27 and 2 + 7 = 9

Seems to work. Like the Amazing Facts of Nine, you might have to add the digits more than once. But you do get to 3, 6, or 9.

This might not help you multiply faster, but it is a quick way to check your work. Say you multiply 51 by 3 and get 151. Well, 1 + 5 + 1 = 7, so you must have made a mistake. The right answer is 153.

Of course, if you mistakenly got 156, those digits add to 12 and then 3. The trick can only show if you're wrong, not prove you're right.

So why do the (Mildly-Startling-If-Not-Amazing) Facts of Three work? Good question. The Amazing Facts of Nine work because 9 is the highest digit in our number system. Basically, the Facts of Three work because 3 is a factor of 9; that is, 3 x 3 = 9. In our number system, there aren't similar Facts of, say, Four or Seven, because those aren't factors of 9. But there are Facts of Three.

See how numbers relate to each other? Here, 9 has a nifty property because it's one less than 10, and 3 has a similar property because it's a factor of 9. Instead of a bewildering horde of digits, you can see patterns. You can multiply 51 x 3 with pencil and paper, then use this trick to check the answer from another angle. Thanks to patterns, the digits of your answer have to add up to three. Math works from every angle. That's what's so cool about it.



Exercise D: 1. What are the Mildly Startling Fact(s) of Three? 2. Do you use this trick to solve a problem or check your work? 3. How might you get “tricked” by this trick? 4. Don't solve these problems. Instead, use this trick and circle which answers

can't be correct. a. 4 x 3 = 13 b. 19 x 3 = 57 c. 27 x 3 = 85 d. 101 x 3 = 303

5. Extra credit. Remember the octal (base 8) number system? Another popular system is hexadecimal (base 16).

1 2 3 4 5 6 7 8 9 A B C D E F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20

Don't let the letters confuse you. Here, A means ten, B means eleven, C means twelve, D means thirteen, E means fourteen, F means fifteen, and 10 means sixteen. 11 means 1 sixteen, 1 one, 12 means 1 sixteen, 2 ones, and so on.

Now, if our base 10 number system has the Amazing Facts of Nine, then hexadecimal (base 16) should have the Amazing Facts of Fifteen, or F. And just as 9 = 3 x 3, in hexadecimal, F = 3 x 5. The factors of F are 3 and 5.

So find a fancy calculator that does hexadecimal. Multiply a few numbers by 3 and add the digits. Do they add up to 3, 6, 9, C, or F? These are the single-digit multiples of 3. Multiply a few numbers by 5 and add the digits. Do they add up to 5, A, or C?

Show your work on the back of this page. Does your research lead to a general conclusion?



Shortcuts and Chunks Multiples of 10 are easy to multiply, aren't they? If you had to do one of these

problems in three seconds, which would you choose? 20 x 7 = ? 19 x 7 = ?

The first, right? Maybe you figured it out just looking at it: 140. You don't have to write it out, it just happens in your head. You forget about the 0 and multiply 2 by 7. Easy. But the second problem is a real problem. Out comes the pencil ... 9 x 7 = 63, carry the 3, 7 x 1 = 7 plus that 3 = 10 ... eventually, bruised by the battle, you reach the answer: 133.

What if you used the easy answer as a shortcut to the tough one? Twenty sevens is 140. Nineteen sevens is one less seven than 140. Right?

Sure. Rather than figuring 19 x 7, you can leap from 20 x 7 to 140. Then go back to 19 x 7 by subtracting 7 from 140.

19 x 7 = ? Ugh. 20 x 7 = 140 This is a close, easier answer. 140 – 7 = 133 Now subtract the extra seven... 19 x 7 = 133 And you're at the right place.

So you can use an easier problem as a shortcut, then add or subtract the difference.

Another trick is to multiply by “chunks.” First multiply the tens, then multiply the ones, then add these sums together. This is all you really do on paper, but you might not realize it.

32 x 8 = ? Yikes. 30 x 8 = 240 Multiply the tens. 2 x 8 = 16 Multiply the ones. 240 + 16 = 256 Add the two answers. 32 x 8 = 256 And we have our final answer.

By breaking a big problem into small chunks, you can do it faster or even in your head. Instead of heaving an unwieldy number like 32, you can toss around three tens and two. Try it.

If you're really ambitious, you can use this on problems like 16 x 24. Multiply 16 x 20, then multiply 16 x 4, then add the answers. No problem.

16 x 20 = 320 16 x 4 = 64 320 + 64 = 384

Hey, if you get fast enough, this could be a real crowd-pleaser.



Exercise E: 1. Write an example that shows how to use an easy multiplication problem as a

shortcut. 2. Write an example that shows how to multiply by chunks. 3. Solve these problems by using one of the tricks you learned in this lesson. Show

your work. a. 49 x 4 b. 56 x 8

4. Solve these problems by using one of the tricks, but do them all in your head. a. 83 x 6 b. 71 x 7 c. 67 x 9 d. 24 x 8 e. 32 x 15 f. 47 x 58



Exercise F: Review 1. Solve these problems by using one of the tricks you learned in this unit. Show

your work. a. 63 x 7 b. 97 x 3 c. 89 x 5 d. 37 x 84

2. Solve these problems by using one of the tricks, but do them all in your head. a. 9 x 8 b. 456 x 10 c. 82.34 x 1000 d. 11 x 5 e. 3 x 9 f. 0.0005 x 10,000 g. 34 x 5 h. 39 x 8 i. 51 x 4 j. 301.03 x 100 k. 7 x 9 l. 78 x 6 m. 93 x 3 n. 8 x 5

3. Don't solve these problems. Instead, circle which answers can't be correct. Explain why.

a. 8 x 5 = 44 b. 9 x 6 = 56 c. 21 x 3 = 61 d. 18 x 5 = 90 e. 9 x 5 = 55



Answer Key: Exercise A

1. You don't need to carry around a calculator./ It's fun/interesting/strengthening. 2. Answers vary. 3. Answers vary. 4. Move each digit once to the left. It works because it simply uses our number

system. When you move digits to the left, you automatically multiply by ten. 5. a. 40

b. 0.04315 c. 40.0 d. 1,392,000

6. Move each digit twice to the left. 7. Move each digit four times to the left. 8. a. 83,100

b. 3.57 c. 0.00091

d. 20.4 e. 4.5309 f. 0.78

Exercise B

1. Subtract 1 from the multiplicand to get the tens digit. Subtract the tens digit from 9 to get the ones digit. Double check by adding the digits together to get 9.

2. a. 36 b. 81 c. 72 d. 45

e. 63 f. 18 g. 27 h. 54

3. a. 9 x 36 = 324 digits add to 9 b. 9 x 252 = 2268 digits add to 18, add to 9 c. 9 x 5,817 = 52,353 digits add to 18, add to 9 d. 9 x 157,931 = 1,421,379 digits add to 27, add to 9 e. 9 x 91,572.425 = 824,151.825 digits add to 36, add to 9

4. (in octal) a. 7 x 2 = 16 1 + 6 = 7

b. 7 x 5 = 43 4 + 3 = 7 c. 7 x 6 = 52 5 + 2 = 7 It seems that this trick works with the highest digit in a number system.

Exercise C

1. To multiply by 5, divide by 2, then multiply by 10. 2. 10 = 2 x 5. If you divide by 2 and then multiply by 10, it's the same as

multiplying by 5. 3. a. 20 b. 60 c. 95 d. 75

e. 35 f. 445 g. 1150 h. 105

4. a. No. An odd number x 5 should end in 5. b. No. An odd number x 5 should end in 5. c. Yes. An even number x 5 should end in 0. b. No. An odd number x 5 should end in 5.



Exercise D 1. If you multiply a number by 3, the digits of the answer add up to 3, 6, or 9. 2. Check your work 3. If your wrong answer is another multiple of 3, the digits will add up to 3 and you'll

be fooled. 4. a and c 5. It seems that in any number system, the factors of the highest digit have a

special property. When you multiply any number times one of these factors and add the digits, the sum is a multiple of that factor. You can add the digits of this sum, and add the digits of that sum, and so on; when the sum is a single digit, it will be a multiple of that factor.

Exercise E Student examples may vary. 1. 19 x 7 = ? 20 x 7 = 140 140 – 7 = 133 19 x 7 = 133 2. 32 x 8 = ? 30 x 8 = 240 2 x 8 = 16 240 + 16 = 256 32 x 8 = 256 3. a. 49 x 4 50 x 4 = 200 200 – 4 = 196

49 x 4 = 196 b. 56 x 8 50 x 8 = 400 6 x 8 = 48 56 x 8 = 448 4. a. 498 b. 497 c. 603 d. 192 e. 480 f. 2726

Exercise F 1. a. 63 x 7 = ?

60 x 7 = 420 3 x 7 = 21 420 + 21 = 441 63 x 7 = 441

b. 97 x 3 = ? 90 x 3 = 270 7 x 3 = 21 270 + 21 = 291 97 x 3 = 291

c. 89 x 5 (Could multiply by chunks, or .) 89 / 2 = 44.5 44.5 x 10 = 445 89 x 5 = 445

d. 37 x 84 30 x 80 = 2400 7 x 80 = 560

30 x 4 = 120 7 x 4 = 28 + 3108

2. a. 72 b. 4,560 c. 82,340 d. 55 e. 27 f. 5 g. 170 h. 312 i. 204 j. 30,103 k. 63 l. 468 m. 279 n. 40

3. a. The answer doesn't end in 5. b. The digits of the answer don't add up to 9. c. The digits of the answer don't add up to 3, 6, or 9.

e. The digits of the answer don't add up to 9.

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Math: Quick Tricks for Multiplication - abcteach.com · Math: Quick Tricks for Multiplication ... Plus: it's weird and cool. Math: Quick Tricks for Multiplication ... you know right