Mastering Multiplication: Tricks, Shortcuts, & Explanations

1. Some Tricks, Shortcuts, and Explanations To Help You Do Multiplication Faster Either On Paper or In Your Head Rick Garlikov

2. It is important to take time and effort to be able to understand some of the tricks and shortcuts here, not just memorize them. In fact, it will be very difficult to remember or be able to use most of them if you don’t understand them as you go. The time and effort will be well worth your while if it helps you understand them. So this slide show is not something meant to be just “viewed” from beginning to end, but for you to take your time and proceed through the slides at your own pace (by clicking on the forward and back arrows at the bottom of the slides) and come back to any slides you don’t understand to work on ‘getting it’ (the slides are numbered on the bottom right so you can note for yourself which ones you need to study further). You should do the same when you read anything, because reading for understanding is usually important, not just reading for speed. It doesn’t do much good to read fast if you aren’t getting much out of what you are reading. It is like taking a test fast or playing sports fast and getting most of the answers wrong or messing up most of the plays because you rushed too much.

3. 1 times any number just gives you that same number, because all it means is you have one of whatever number you started with. So 1 time 5 is just 5. 1 times 8 is just 8. 1 times 37 is 37. 1 times 5,736 is 5,736. 1 times 1,023,749,216 is 1,023,749,216 and 1 times 1 = 1

4. Multiplying any number is like counting by that number. So if you multiply 2 by some number it is like counting by two’s that many times. It is like counting socks two at a time in pairs. If you have 6 pairs of socks, that is “two, four, six, eight, ten, twelve” socks. 2x6 is 12. Multiplying by 3 is like counting by three’s: “three, six, nine, twelve, fifteen, eighteen, twenty one”, etc. Multiplying by 4’s is counting by four’s: “four, eight, twelve, sixteen, twenty, twenty four, twenty eight, thirty two”, etc. Multiplying by 5’s is like counting by five’s: five, ten, fifteen, twenty, twenty five, thirty, thirty five, etc. Multiplying by 10’s is like counting by ten’s: ten, twenty, thirty, forty, fifty, sixty, etc. And multiplying by 10 is easy, since you just put a zero on the end of the number you are multiplying by ten. If you have ten times two, that is 20, ten times three is 30. If you have ten times four, that is 14 tens and you count up by 10 fourteen times, you get to 140. Five hundred tens would be 5000. Ten times 2 million is 20 million. Or in numerical form: 10 x 2,000,000 = 20,000,000.

5. In the next slide is a 15 x 20 multiplication table where the numbers in the boxes are the products of the numbers in the top (blue) row of the column and the numbers in the left hand (red) column in the same row. (Remember, the answer or result you get whenever you multiply numbers together is called the “product”, just like the answer you get in adding numbers is called the “sum”.) Notice that in each column, as you from the top downward, you are counting by the top number (in the blue row). And it is the same for each row as you go from left to right, where you are counting by the number in the first column (the red column). The pattern in each row is identical to the pattern in the column that begins with the same number. The diagonal green rectangle shows the pattern of the products of numbers multiplied by themselves – that is, the numbers “squared” or forming a square of equal sides whose length is the number that is being squared in whatever unit of measure, such as feet, inches, centimeters, seconds, hours, etc.

7. Here is a 15 x 15 table with the cells being more square in shape themselves. The numbers multiplied by themselves are in the diagonal green rectangle. Multiplying the left most number (red column) by the same number in the top (blue) takes you stepwise over one and down one as you go.

8. And since multiplying any number by five gives you half the amount that multiplying it by ten does, it is often easy to multiply by five, since you just take half of the number with the zero added to it from multiplying it by 10. So 5 x 40 = half of 400, which is 200. 5 x 50 = half of 500, which is 250. 5 x 25 = half of 250, which is 125. 5 x 18 = half of 180, which is 90. 5 x 844 = half of 8,440, which is 4,220.

9. The other thing to keep in mind about multiplying by 5 is that the answer will either end in a five or a zero, since when you count by 5’s every number you say will either be a multiple of ten or will be the five number in between the last ten and the next ten. And moreover, if you multiply 5 (or any number ending in 5) by an even number, it will end in a zero, and if you multiply 5 (or any number ending in 5) by an odd number, it will end in a five. For example, 15 x 2 = 30; 15 x 3 = 45 (which is 15 more than 15 x 2); 15 x 4 is 60 (which is double 15 x 2, since multiplying anything by 4 gives you double what multiplying it by 2 does, since 4 is twice as much as 2). Other examples of multiplying five by odd or even numbers are: 2 fives are 10, 5 fives are 25, 6 fives are 30, 7 fives are 35. Notice, if you go down the 5 column, or from left to right in the 5 row, in the multiplication chart, the numbers alternate between ending in 5 and ending in 0. And notice that the ones ending in zero will be the 5 multiplied by an even number and the ones ending in 5 will all be the 5 multiplied by an odd number. That doesn’t prove even number multiples of 5 and of numbers ending in 5 will always end in 0, or that odd number multiples of them will end in 5, but it illustrates it.

10. Whenever you multiply an even number by anything, you will end up with an even number, because every even number is a pair, and no matter how many pairs you have, they will always be able to be evenly divided (down the middle of the pairs). • For example if you start out with 17 and you multiply it by 2, you will then have a pair of 17’s, which will have to be an even number because when it is divided by 2, it will give the whole number 17 that you started with. That will be true for any number you multiply by 2 – dividing the product in half will give you the whole number you started with.

11. That means when you multiply even numbers together or when you multiply odd numbers by an even number, you will still have an even number of things. • That is not only because again you are going to get pairs of things by multiplying by an even number, but also for another reason:

12. Here is another reason why multiplying an odd number by an even number gives you and even number: • Since every odd number is one more than the even number before it, when you double an odd number, such as 27 (which is 26 plus 1), you are adding another 26 plus 1 to it. The two 26’s will still be even (52) but now the 1 will be doubled also (two 2), and the answer will be 54.

13. Whenever you multiply two odd numbers together, you will get an odd number, because you will have an odd number of odd numbers. • So suppose you multiply 11 x 17. That will be 1 more 17 than 10 x 17. Ten times 17 will be 170, which is an even number, and since you need to add one more 17 to it, the answer will be an odd number, 187. • When you multiply any odd number by an odd number, that is the same thing as multiplying it by the even number that is one less, which will give you an even number, and then you still have to add the odd number back in one more time. In other words, suppose you are multiplying 61 x 33. That will be the same as 60 x 33 plus 1 more 33. Since 60 is an even number, 60 x 33 will be an even number, and adding the odd number 33 to it, will give you an odd number.

14. This trick shortcut is really cool: To multiply any number ending in 5 times itself (meaning to ‘square any number ending in 5’), you multiply whatever number is in front of the 5 by the next higher number than it, and then you stick 25 on the end of what you get. For example 15 x 15 = 225 because you multiply the 1 by the next higher number, 2, and you get 2, which becomes 225 when you stick the 25 on after it. 35 x 35 is easy: multiply the 3 times the next higher number, 4, to get 12, and put the 25 on the end, to give you 1225. 105 x 105 = 10 x 11, which is 110, with 25 stuck on the end, to give 11,025. Check these out with a calculator. Try some other numbers ending in 5 multiplied times themselves, that is, ‘squaring’ them.

15. This trick shortcut is really cool but will take some particular time and effort to understand. But it will be well worth that time and effort once you do understand what it means and how it works. • There is an easy way to multiply some numbers that are a convenient and equal distance apart from a number halfway between them, such as 14 x 26, numbers which are each 6 away from 20. 14 = 20 – 6, and 26 = 20 + 6. And whenever you multiply two sets of numbers like that together, the answer will be the first number (in this case the 20) times itself minus the second number (in this case the 6) times itself. 20 x 20 is 400 and 6 x 6 = 36. And since 400 - 36 = 364, 14 x 26 = 364. Check it out on the calculator. The general rule for this, if we represent numbers by letters is: • (a + b) times (a – b) = a²– b² • where the symbol “²” is a shortcut means of saying to multiply the number in front of it times itself. • So another example of that would be 38 x 32, which is (35 + 3) times (35 – 3). The answer will be 35² - 3², which is (35 x 35) – (3 x 3). You saw before that it is easy to multiply a number ending in 5 times itself, so 35 x 35 = 1225, and 3 x 3 is 9. So the answer is 1,225 – 9 = 1,216, which a calculator will show is correct.

16. Now I want to point out one really easy, but kind of strange, thing about multiplication. The easy part is that: multiplying any number by zero = zero So 24 x 0 = 0; 427 x 0 = 0; a kabillion x 0 = zero; 1 x 0 = 0. That makes perfectly good sense when you think of it one way, but not if you think about the other way: It makes sense when you think of it as counting up from nothing by adding nothing to it. No matter how many times you add nothing to nothing, whether a 100 times or a billion times, you still have nothing. So 20 x 0 is still going to be zero; and 28 x 0 = 0; and 432,000,251 x 0 = 0. However, it seems odd that if you start with something like 7, and multiply it by zero, which sounds like not multiplying it at all, that you would lose the 7 you had, and have zero. And it sounds like adding 7 to 7 no times at all would also eliminate the 7 you are starting with, which seems strange and like it shouldn’t be true. So that is one rule that only makes sense in one way of thinking about it, so this is one rule that may be an exception to trying to understand the ideas involved, and instead, just remember: multiplying any number by zero = zero

17. Multiplying by 9I don’t know why this one works. I haven’t figured out a way to prove it or to “see” it yet. When you multiply any number by 9, the individual numerals in the answer will add up to 9 or to a multiple of 9 whose digits will add up to 9 (or a multiple of 9, whose digits will add up to 9 or a multiple of 9…etc). And if you are multiplying 9 times any single digit number, the numeral on the left side of the answer will be one less than the number you are multiplying by 9, and the numeral on the right will be whatever you need to add to the numeral on the left in order to get 9. For example 9 x 3 = 27. The “2” in 27 is one less than the 3 you were multiplying the 9 by, and the “7” in 27 is what you need to add to the “2” in order to get 9. Another example is 9 x 8. One less than 8 is 7, so the first numeral in the answer will be “7”, and then the next numeral will be what you need to add to 7 to get 9, which, of course is 2. So the product of 9 x 8 = 72. If you are multiplying a large number by 9, you will get a product whose numerals will sum to a multidigit number, but that sum will be a multiple of 9 and its numerals will either add up to 9 or to another multiple of 9, whose digits will add up to 9 or a multiple of 9, whose digits will …. Etc. For example, using a calculator: 83,657,822 x 9 = 752,920,398 -- whose digits add up to 45, whose digits add up to 9.

18. In a minute I will show you the coolest trick of all in multiplication, but first I need to show you why you can multiply two numbers in either order, so that, for example, 9 x 3 = 3 x 9. Although this may seem to be obviously fine, it is not obvious once you realize it means something like “3 containers with 9 things in each them is the same amount of those things as 9 container with 3 things in each of them.” Clearly those are two totally different conditions. If, for example, you have to make up little gift bags for a party for 12 people, where you give everyone five different candy bars, you need 12 bags with five candy bars each in them, not 5 bags with 12 candy bars in each of them, even though it is 60 candy bars total, no matter how you divide them up. But if you only have 5 bags with 12 candy bars each, seven of your friends will be upset and five of your friends will get tummy aches. So how can it be that 12 x 5 = 5 x 12? And, in general, if we let letters stand for any possible numbers, the question is how can it be that: (A times B) = (B times A) and therefore that A bags with B things each in them will be the same number of things as B bags with A things in each of them. Before moving on to the next slide, see whether you can figure out how or why that is.

19. The reason it is, can be seen by considering a rectangle, like the following divided into columns and rows: It has three columns and eight rows to give it 24 boxes.

20. If we turn it on its side, we will have Now it has 8 columns and 3 rows instead of 3 columns and 8 rows, but still the same 24 boxes because it is the same column of boxes we started with. So if we think of the bags of candy as represented by the columns, there will be 8 bags with 3 pieces each in the horizontal rectangle, and there will be 3 bags with 8 pieces of candy each in the vertical one. It doesn’t matter whether you are multiplying the number of boxes per column times the number of columns, to get the total number, or multiplying the number of boxes per row times the number of rows to get it. The answer is the same because the number of boxes is the same. How you divide them or arrange them doesn’t change that.

21. This brings me to one of the really coolest tricks of all, but it will take some explaining that you will have to think about as I show it all to you in the next few slides.

22. *************…**** Here are a number of * symbols all in a row. The three periods in between the two groups means “there are more *’s, but we don’t know how many more, and it could be any number, even a negative number that subtracts some of the ones we see”. So in this whole row of * symbols there could be 50 or just 1, or even none, altogether, or 172, or a million, or a gazillion. Whatever the number of them is, we will call it “a”. So “a” stands for some number, but we don’t know exactly what number and it could be any number. If someone asks how many *’s there are, we would say “There are “a” of them.”

23. Now we will make the same thing, except in a column instead of a row. And again, the three dots means there are more, but it could be any number. Before we said there were “a” *’s in the row. Now we will say there are “b” *’s in this column. “b” could be the same number as “a” or it could be different. It doesn’t matter. *****…*******

24. Now let’s combine the two sets of *s, to make “a” columns and “b” rows: *************…**** *************…**** *************…**** *************…**** *************…**** . . . *************…**** *************…**** *************…**** *************…**** *************…**** *************…**** *************…**** “b” rows “a” columns And that means the total number of *’s is “a” multiplied by “b”. I am going to write that as “ab”, just for a shorthand way of saying “a times b”.

25. Now consider the following large rectangle made up of four smaller rectangles:

27. There are *’s in each of the color boxes and if you add the *’s in the red box, the blue box, the yellow box, and the green box together, you get the number of *’s that are in the large box that has the black border.

28. So, for example, if there are 700 *’s in the red box, 280 in the blue box, 800 in the yellow box, and 320 in the green box, then there will be 700 + 280 + 800 + 320, which is 2100 altogether in the whole big rectangle.

29. Now notice, the number of *’s in each of the color boxes is the number of rows in that color box multiplied by the number of *’s in each row in that same color box. And notice also that the way I have drawn this, the red box and the yellow box have the same number of columns as each other. Also, the blue box and the green box have the same number of columns as each other. And notice that the red box and blue boxes have the same number of rows as each other, and the yellow box and green box have the same number of rows as each other too. This is going somewhere really, really cool about a trick for doing lots of multiplications fast and easy, so just stay with me a little longer.

30. So, for example, let’s say the *’s symbols and the … symbols represent that there are 8 columns in the red and yellow boxes and four columns in the blue and green boxes. And let’s say there are 15 rows in the red and blue boxes, and 20 rows in the yellow and green boxes. That means there are 8 + 4 = 12 columns altogether in the big rectangle, and there are 15 + 20 = 35 rows altogether in the large rectangle. Multiplying 12 times 35 on the calculator = 420 *’s in the large rectangle. So let’s see how many are in the different color rectangles and then add them together to see whether we get 420 or not. Red = 8 x 15 = 120 Yellow = 8 x 20 = 160 Blue = 4 x 15 = 60 Green = 4 x 20 = 80 120 + 160 + 60 + 80 = 420 And since this works this way, there is a trick for doing multiplication of numbers we can group like the rectangles. 15 rows 20 rows 4 columns 8 columns

31. Red box = “a times c” (written as “ac”) Blue box = “a times d” (written as “ad”) Yellow box = “b times c” (written as bc)Green box = “b time d” (written as bd) Whole box = (a + b) times (c + d) written as (a + b)(c + d) And since the whole box = red box + blue box + yellow box + green box, then (a + b)(c + d) = (ac + ad + bc + bd) which is easier to remember than it looks because it is formed by multiplying each of the parts in the first parentheses times each of the parts in the second parentheses, like this: (a + b)(c + d) = (ac + ad + bc + bd) a rows b rows which is the “a” times the “c” first, and then times the “d”and then the “b” times the “c” first and then times the “d”. d columns c columns

32. What This Means for Multiplying Numbers Suppose we want to multiply 28 times 56. 28 times 56 is the same as (20 + 8) times (50 + 6). And since the formula before is shown to be (a + b)(c + d) = (ac + ad + bc + bd), that means (20 + 8) times (50 + 6) = (20 times 50) + (20 times 6) + (8 times 50) + (8 times 6) = (1000) + (120) + (400) + (48) = 1,568

33. We could do this another way too, since: 28 x 56 also is (30 – 2) times (60 – 4), written as (30 – 2)(60 – 4) Doing this with subtraction is a little more difficult to show with rectangles, but we can do it in the following way:

34. Suppose we want to know how many things are inside the white rectangle. I have drawn the columns in the white rectangle so that they equal the number of columns in the red rectangle minus the number of columns in the blue rectangle. And the number of rows in the white rectangle is the number of rows in the red rectangle minus the number of rows in the yellow or green rectangles (but not both). So the white rectangle’s dimensions are (a – b) rows, and (c – d) columns. So the number of things in the white rectangle = (a – b) times (c – d), which in shorthand we can write as (a – b)(c – d). a rows b rows d columns c columns

35. Now, notice: the number of things in the white rectangle will equal the number of things in the red rectangle (which is a times c) minus the number of things in the blue rectangle (which is a times d) and the number of things in the yellow rectangle (which is b times c), except that you have to add back in the number of things in the green rectangle, because otherwise we subtracted them twice by subtracting the rows in the yellow rectangle and the columns of the blue rectangle. The purple rectangle is the same size as the green one, an we would have removed that area twice by subtracting both the yellow rows and the blue columns from the red rectangle. So we are putting one of those subtractions back by adding (b times d). So, the number of things in the white rectangle is the number of things in the: Red, minus blue and also minus yellow, but plus green, which is: (a – b)(c – d)) = [(ac) – (ad) – (bc) + (bd)] a rows b rows d columns c columns

36. So then, going back to 28 times 56, thought of as: (30 – 2)(60 – 4) that will be equal to [(30)(60) – (30)(4) – (2)(60) + (2)(4)] which is: [1800 – 120 – 120 + 8] = 1,568 which is the same answer as before. Notice that when we do this strictly in numbers instead of by the rectangles, (30 – 2)(60 – 4) will be [(30)(60) + (30)(-4) + (-2)(60) + (-4)(-2)] which by the way multiplication with positive and negative numbers works ends up being the same as [(1800) + (-120) + (-120) + (+8)] =[(1800) - (120) - (120) + (8)] = 1,568 (30 – 2)(60 – 4) a rows b rows d columns

37. Understanding (a + b)(c + d) = (ac + ad + bc + bd) which is the “a” times the “c” first, and then times the “d”and then the “b” times the “c” first and then times the “d” will show why slide the general rule in slide 15 works, that (a + b)(a – b) = a² – b² since it will be: [(a)(a) + (a)(-b) + (b)(a) – (b)(b)] = a² - ab + ba - b² = a² - b²

38. And it also gives a formula for squaring any number, such as 87, which is (80 + 7)²: The square of the sum of any two numbers x and y is represented by: (x + y)2 which is always equal to (x2 + 2xy + y2) because (x + y)2 is the same as (x + y)(x + y), which, by our equation for multiplying gives us (x)(x) + (x)(y) + (y)(x) + (y)(y), which, since (x)(y) is equivalent to (y)(x), gives: x2 + 2xy + y2 So, to square a number like 87, we would get (80) 2+ 2(80)(7) + (7)2 = 6,400 + 2(560) + 49 = 6,400 + 1,120 + 49 = 7,569, which you can see is correct if you check it with a calculator. Now, of course it is faster to do that with a calculator, but you don’t always have a calculator handy, and more important, you can do this in algebra with variables instead of numbers, and you can’t multiply variables on a standard, simple calculator.

39. If you would like to see and work through why squaring numbers that end in 5 follows the trick in slide 14, see www.garlikov.com/math/SquaringNumbersEndingIn5.html