101 Shortcuts in Math2

7/29/2019 101 Shortcuts in Math2

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. Y e > ~ Prillling /975

Copyright 1965 by Gordon RockmakerAll r i @ h t ~ re'>ened :"o part of thts work co,crcd the copyri)!Cht hereon may bereproduced or u'ed m any form ot b' an) mcan,-grapht=. c!c:tronic, or mechanical.including p h o t o c o p ) m ~ rccordinA taping. or ~ l f o t m a t i o n 'lotagc and retncval S)stems-without permi,ston of t11c p u h l t ~ h c rFo r mformalion addrtss:F redencl FeU Publi,hel"o. Inc.386 Part.. A'enuc


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vi PREFACEfractions, mixed numbers, and percentage. In a word,they range across the whole field of calculation one 1slikely to use.In compiling the short cuts to be included in this book,only authentic ones were chosen. An authentic short cu tis one that will produce an answer quickly and easily with-out the necessity of going through the usual intermediatesteps, and it is usually very specific. By cutting throughthe time-consuming mechanical operations and gomgstraight to the heart of the answer, a tremendous amountof needless work is avoided.All computations in this book ar e performed from leftto right. This is the first time this approach has beenapplied to short-cut methods . It permits you to write theanswer to a problem immediately in the same sequence inwhich it is read-from left to right.Emphasis has been deliberately placed on general usesrather than on the specihc uses of a particular short cut.The practical value of a short cu t is that i t can be used ina wide variety of applicahons. By demonstrating only oneor at best a few specific applications, the danger existsthat the reader will not venture beyond the ones described.

No book can contain every short cut in math, but thisbook does include some of the most useful modern methodsdevised.

Perhaps the most important function of this book is tointroduce you to the wide practical application of mathematical short cuts . Using your own creative spirit andthe curiosity to experiment, there is no limit to the number of short cuts you can devise for your own specialneeds.

TABLE OF CONTENTS

PREFACEINTRODUCTION

Chapter 1SHORT CUTS IN ADDITION

1. Adding Consecutive Numbers2. Adding Consecutive Numbers Starting from l3. Finding the Sum of All Odd Numbers Starting from 14. Finding the Sum of All Even Numbers Starting from 25. Adding a Series of Numbers With a Common Difference6. Adding a Series of Numbers Having a Common Ratio

Chapter 2SHORT CUTS IN MULTIPLICATION

THE DIGITS7. Multiplying by Numbers Ending in Zeros8. :\lultiplying by 29. :\1ultiplying by 310. Multiplying by 4

11 . Multiplying by 512. Multiplying by 613. :\iultiplying by 714. Multiplying by 815. Multiplying by 9

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vl

7

91011121314

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17202124272931333537


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NUMBERS BEGINNING OR ENDING IN I 40 41. Multiplying Two Two- Dig1t !'umbers Ending in 9and Whose Tens D1gits Add to 10 7816. Multiplying by 11 41 Multiplying by a Two- Digit Multiple of 9 7917. Multiplying by 12 43 42.43. Multiplying by Any Two-Digit Number Ending in 9 8018. Multiplying by 111 4519. Multiplying by a Multiple of 11 47 SQUARING NUMBERS 8220. Multiplying by 21 4821. Multiplying by 121 50 44. Squaring Any Number Ending in 1 8322. Multiplying by 101 51 45. Squaring Any Two - Digit Number Ending in 5 8423. Multiplying by 1,001 52 46. Squaring Any Number Ending in 5 8524. Multiplying by One More Than a Power of 10 53 47. Squaring Any Three-Dig1 Number Ending in 25 8625. Multiplying Teen" Numbers 55 48. Squaring Any Four-Digit Number Ending in 25 8826. Multiplying by Any Two-Digit Number Ending in 1 56 49. Squaring Any Two- Digit Number \\.'hose Tens Digit is 5 9250. Squaring Any Number Ending in 9 93

NUMBERS BEGINNING OR ENDING IN 5 59 51. Squaring Any Number Consisting Only of Nines 9427. Multiplying by 15 60 52. Squaring Any Two-D1git Number 9528. Multiplying by 25 62 MULTIPLYING TWO NUMBERS THAT DIFFER29. Multiplying by 52 63 ONLY SLIGHTLY 9830. Multiplying a Two- Digit Number by 95 6431. Multiplying by 125 65 53. Multiplying Two Numbers Whose Difference Is 2 9932. Multiplying Two Two-Digit Numbers When Both 54. Multiplying Two Numbers Whose Difference Is 3 100

End in 5 and One Tens Digit Is Odd While th e 55. Multiplying Two Numbers Whose Difference Is 4 101Other Is Even 66 56. Multiplying Two Numbers Whose Difference Is 6 10233. Multiplying Two Two- Digit Numbers When Both 57. Multiplying Two Numbers Whose Difference Is AnyEnd in 5 and Their Tens Digits Are Either Both Small Even Number 103Odd or Both Even 6734. Multiplying Two Two- Digit Numbers Whose Tens MORE SHORT CUTS IN MULTIPLICATION 105Digits Ar e Both 5 and Whose Units Digits Ar e 58. Multiplying Two Two-Digit Numbers Whose TensBoth Odd or Both Even 68

35. Multiplying Two Two-Digit Numbers Whose Tens Digits Ar e the Same 106Digits Ar e Both 5 and One Units Digit is Odd 59. Multiplying Two Two-Digit Numbers Whose UnitsDigits Are the Same 107While th e Other is Even 69 60. Multiplying Two Numbers That Ar e Just a Little36. Multiplying Two Two-Digit Numbers Whose Tens Less than 100 109Digits Ar e Both 5 and Whose Units Digits Add 61. Multiplying Two Numbers That Ar e Just a Littleto 10 70 Less than 1,000 111

NUMBERS BEGINNING OR ENDING IN 9 73 62. Multiplying Two Numbers That Are Just a LittleMore than 100 11337. Multiplying by 19 74 63. Multiplying Two Numbers That Ar e Just a Little38. Multiplying by 99 75 More than 1,000 11539. Multiplying by 999 76 64. Multiplying Two Numbers Whose Units Digits Add to40. Multiplying by a Number Consisting Only of Nines 77 10 and th e Other Corresponding Digits Ar e Equal 117viii ix


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~ ~ l l l i i i ' I Z O ~ I I I I I i E i 1 l ~ i l l i l i J I I m i : I : Z i i i i i O I I I ~ - - - - - - - - ~ - - ~ . . .

Chapter 5Chapter 3

SHORT CUTS IN SUBTRACTION 119 SHORT CUTS WITH FRACTIONS, MIXEDNUMBERS, AND PERCENTAGE 15365 . Subtracting a Number from the Next Highest Power

of 10 12066 . Subtracting a Number from Any Power of 10 121 83 . Adding Two Fractions Whose Numerators Ar e Both 1 15484 . Finding the Difference Between Two Fracttons WhoseNumerators Are Both 1 155Chapter 4 85 . Multiplying by 3/ 4 156

SHORT CUTS IN DIVISION 86 . Multiplying by 2-1/2 157123 87 . Multiplying by 7-1/2 15988 . Multiplying by 12-1/2 160

DETERMINING A NUMBER'S DI VISORS 123 89 .Multiplying Two Mixed Numbers Whose Whole Num-

bers Are the Same and Whose Fractions Add to 1 16167 . Divisibility by 2 125 90 . Multiplying Two Mixed Numbers When the Difference68 . Divisibllity by 3 126 Between the Whole Numbers Is 1 and the Sum of69 . Divisibility by 4 127 the Fractions Is 1 16270. Divisibility by 5 128 91 . Squaring a Number Ending in 1/ 2

16371 . Divisibility by 6 129 92 . Dividing by 2-1/2 16472. Divisibility by 7 130 93 . Dividing by 12-1/2 16573 . Divisibility by 8 132 94 . Dividing by 33-1/3 16674. Divisibility by 9 133 95 . Finding 16-2/3G1 of a Number 16775. Divisibility by 11 134 96 . Finding 33-1/3 1: of a Number 16876. Divisibility by 13 135 97 . Finding 37-1/2 % of a Number 16998 . Finding 62-1/2 % of a Number 170

NUMBERS ENDING IN 5 137 99 . Finding 66-2/3 l7 of a Number 17177. Dividing by 5 138 100.

Finding 87-1 /2 1{ of a Number 17278. Dividing by 15 13979. Dividing by 25 14280 . Dividing by 125 143 Chapter 6

MORE SHORT CUTS IN DIVISION 145 POSTSCRIPT 17581. Dividing by 9 14682 . Dividing by Factors 149 101. Do-It-Yourself Short Cuts

176

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INTRODUCTIONCUTTIKG C O R ~ E R S

Whether due to curiosity or sheer laziness, man hasalways been experimenting, searching for and stumbling upon ways of making work easier for himself. That anonymous caveman who chipped the corners off a flat rock andinvented the wheel started this tradition.

Most of man's efforts in the past were directed at conserving or increasing his muscle power, but as time wenton some were aimed at saving wear and tear on anothervital organ: his brain. It followed naturally that his attention turned to reducing such laborious tasks as calculating.

WHAT SHORT CUTS AREShort cuts in mathematics ar e ingenious little tricks in

calculating that ca n save enormous amounts of time andlabor - not to mention paper - in solving otherwise complicated problems There are no mag1cal powers connectedwith these tricks; each is based on sound mathematicalprinciples growing out of the very properties of numbersthemselves. The results they produce are absolutely ac curate and infallible when applied correctly. Short-cutmethods ar e by no means of recent origin; they were knowneven to the ancient Greeks. The supply of short cuts is unlimited. Many are known, and many are yet to be discovered. Th e 101 short cuts included in this book have beenselected because they are easy to learn, simple to use, andcan be applied to the widest range of calculating problems.

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2 INTRODUCTION

PUTTING NUMBERS IN THEIR PLACETh e numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 are called di gits .

Integers ar e numbers consisting of one or mor e di gits . F orexample, 72,958 is an integer consisting of five digit s , 7,2, 9, 5, and 8. In practice, the word number is applied tomany different combinations of digits ranging from wholenumbers, to fra ctions, mixed numbers , and decimals. Th eword integer, however, applies only to whole numbers.

Each digit m a number ha s a name based on its positionin the number. The number system we are accustomed todealing with is based on the number 10. Each number position in this system is named fo r a power of 10. The position immediately to the left of the decimal point of anumber is called the units position. In the number 1.4 thedigit 1 is in the units position an d is called the units digit.In fact, an y digit that occupies that position is called theunits digit. Th e next position to the left of the units position is called the tens position, and any digit occupying thatspace is called the tens digit. In the number 51.4 the 5 isthe tens digit. Continuing to the left, in order, are thehundreds, thousands , ten-thousands , hundred-thousands,milhons positions, and so on.

The positions of the d1gits to the right of th e dec1malpoint also have names similar to those to the left. Th eposition immed1ately to th e right of the decimal pomt 1scalled the tenths position. Notice that the name is tenthsan d not tens. In fact, al l positions to the nght of the decimal point end in ths. Th e next position to the right of thetenths position is the hundredths position, then the thousandths position, and, in order, the t e n - t h o u s a n d t h s ~hundred-thousandths, the millionths.

INTRODUCTION 3Decimal Point

Units----"'"'T e n s - - - - ~Hund r ed s - -----.Thousands - - " ' " 'T e n - t h o u s a n d s ~

~ ~ ~ ~ ~ ~ - t h a u s a n ~8,367,351.42

.----- - Tenthsr----Hundredthsr -- - Thousandthsr-Ten-thousandths

Hundred-thousandths.11 MillionthsRemember, the position names never change. Th e positionto the left of the decimal point is always the units position;the on e to the right is always the tenths position, no matterwhat digit occupies the space.

In addition to the names of the posit ions as given above,the letters A, B, C, ... will be used in this book to help ex plain the various short-cut methods. Thus, in some shortcuts the digits will be arranged as given below:

ABCDEFGH I JKLM

8 3 6 7 3 5 1.4 2 8 0 3 9Th e letters themselves have no significance beyond helpingidentify and locate a particular digit under discussion inthe short cut. Fo r that reason it is important not only tolearn the various position names but also to gain familiarity with the letter notation just mentioned. Both will be usedfrequently throughout this book.

GETTING THE POINTAll numbers may be considered to have a decimalpoint.

The point is used to separate those numbers that are equalto or greater than 1 from those numbers that are less than1. Even i f we write a number without a decimal point, it


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4 INTRODUCTIONis understood that there is one to the right of the unitsdigit. Fo r example, we ca n write seven dollars and fortynine cents as

$7.49Clearly the decimal point separates the dollars figure (1 ormore) from the cents figure (the part that is less than onedollar). But when we speak of seven dollars alone we maywrite it as $7 or $7. or $7.00. These three forms are ex actly equal. In the first case th e decimal is omitted butnevertheless is understood to be to the right of the 7. It isalso understood that the only digits that ca n be placed tothe right of the decimal point without changing th e value ofthe number are zeros. And as many zeros may be placedto the right of the decimal point as we wish. Later in theapplication of many short-cut methods you will se e whythis is an important property of decimals.

LEARNING TO TAKE THE SHORT CUTTh e preceding sections dealt with the language of mathematics . Before studying any of the methods that follow,

make certain that you are thoroughly familiar with theterms that will be used. When you read about the "hundredsdigit," you must immediately recognize that this refers toa position in an integer and not the number 100. Also,never confuse the hundreds digit with the hundredths digit.Once you have familiarized yourself with th e language,the next step is to develop a routine fo r learning and mem

orizing the short cuts. Maximum efficiency ca n be achievedonly through constant practice. You will soon discoverthat short cuts fall into logical groups or classifications.Short cuts involving numbers ending in 5 are an exampleof such a group. Learn to recognize a problem in termsof it s group. I t would be pointless to have to refer to thisbook each time you wanted to apply a short cut.

INTRODUCTION

TAKING THE SHORT CUT FROM LEFT TO RIGHTMost of us were taught the arithmetic operations of

multiplication, addition, and subtraction from right to left .We always started from the units digit and worked to theleft. After we got ou r answer, we reversed the number inour mind and read it from left to right. Not only wa s theprocess awkward, but the mental gymnastics wasted time.Take this simple example:

364 X 7Th e product was obtained in the following order:

8, 4, 5, 2Then to read th e answer, it became

2,548

5

Why cannot answers be obtained in their natural readingorder? There is no reason at all why we cannot solve prob -lems just as easily from left to nght as we do from rightto left.In this book all work will be performed in the naturalorder in which we write and read numbers - from left toright. Initially this method may seem strange; but oncemastered, it s advantages will become evident and the t imesaving ease with which it ca n be used will prove it s worth.

In this book, the term "first digit" refers to the lefth a ! l ~ m o i t .c!!Jii!_.

FOUR TO GOHere are a few hints to ge t you started on the right foot.First, read and reread the Rules as many times as necessary (at least twice) until a general idea of the short-cut

method is established in your thought. Keep in mind that

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6 INTRODUCTIONyou ar e studying and not reading a novel. Tr y to followthe method in general terms without thinking of specificnumbers at this stage.

Next, follow the sample problems carefully, step by step.Do not skip steps just because you feel they involve sometrivial operation, such as adding 1. After you have readthe sample problem a few times, try to do the same prob-lem yourself, writing the numbers as you go along . Do notrefer to the book at this point. I f you don' t get al l the stepscorrect, go back over them again. You may have to re -read the Rule .

Finally, when you ar e completely satisfied that you havemastered the short cut, tr y the Practice Exercises. Theanswers shoulo be written directly in the space provided.Tr y doing intermediate steps mentally. Very soon you' llfind that you can solve most problems without paper andpencil.

Remember, systematic study and concentration on whatyou ar e doing ar e vital to the mastery of each of the 101short cuts in mathematics.

Chapter 1SHORT CUTS IN ADDITION

Addition is probably the first arithmetic operation mostof us learned after we found out what numbers were. Doyou remember the admonition, never to add dissimilar ob-jects? One must not add 2 oranges to 2 apples (unless onewere making fruit salad). Different methods of addingwereusually taught to help speed the process. However, strictlyspeaking, there ar e no short cuts to adding random groupsof numbers. No matter what method of addition is used,eventually they al l require adding digit by digit until thefinal sum is obtained.In adding regular sequences of numbers, short cuts ar epossible. These sequences can be groups of consecutivenumbers, series of numbers that differ by some constantamount, or series of numbers where each term differsfrom the preceding term by some common ratio. An ex -ample of the first group would be the numbers

73, 74, 75, 76 , 77 , 78, 79, 80, 81This is a series of consecutive numbers from 73 to 81. Anexample of the second series would be the numbers

5, 12, 19, 26, 33In this series each number is always 7 more than the pre-ceding number. An example of the third group would bethe series

7, 21, 63, 189, 567Here each number is 3 times more than the precedingnumber. - -

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8 SHORT CUTS IN ADDITIONIn each case, of course, the sum of the terms in theseries can be found by simply adding digit by digit, but

briefer, less laborious ways of finding these sums ar e pre-sented in the short cuts that follow. 1

ADDING C O ~ S E C U T I V E NUMBERSRule: Add the smallest number in the group to thelargest number in the group, multiply the re -

sult by the amount of numbers in the group,and divide the resulting product by 2.

Suppose we want to find the sum of al l numbers from33 to 41. First, add the smallest number to the largestnumber.

33 + 41 = 74Since there ar e nine numbers from 33 to 41, the next stepis

74 x 9 = 666 (see Short Cut 15)Finally, divide the result by 2.

666 + 2 =- 333 AnswerThe sum of al l numbers from 33 to 41 is therefore 333.

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2ADDING CONSECUTIVE NUMBERS STARTING FROM 1

Consider the problem of adding a group of consectivenumbers such as: 1, 2, 3, 4, 5, 6, 7, 8, an d 9. How wouldyou go about finding the1r sum? This group IS ce rtainlyeasy enough to add the usual way. But i f you're reallyclever you might notice that the first number, 1, added tothe last number, 9, totals 10 and the second number, 2, plusth e next to last number, 8, also totals 10. In fact, startingfrom both ends and adding pairs, the total in each case is1 0. We find there ar e four pairs, each adding to 1 0; thereis no pair for the number 5. Thus 4 x 10 = 40; 40 + 5 =45. Going a step further, we can develop a method fo rfinding the sum of as many numbers in a row as we please.

Rule: Multiply the amount of numbers in the groupby one more than their number, and divideby 2.As an example, suppose we are asked to find the su m of

al l the numbers from 1 to 99. There are 99 integers inthis ser ies; one more than this is 100. Thus

99 X 100 = 9,9009, 900 + 2 4, 950 Answer

Th e su m of al l numbers from 1 to 99 is therefore 4, 950.

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3FINDING THE SUM OF ALL ODD NUMBERSSTARTING FROM 1

Rule: Square the amount of numbers in the series.To show this, the su m of al l numbers from 1 to 100

will be calculated . There are 50 odd numbers in this group.Therefore

50 x 50 = 2, 500 AnstverThis is the su m of al l odd numbers from 1 to 100. As acheck, we ca n compare this answer with the answers foundin Short Cuts 2 and 4.

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4FINDING THE SUM OF ALL EVEN NUMBERSSTARTING FROM 2

Rule: Multiply the amount of numbers in the groupby one more than their numberWe shall us e this rule to find the sum of al l even num

bers from 1 to 100. Half of the numbers will be even andhalf will be odd, which means there ar e 50 even numbersfrom 1 to 100. Applying the rule,

50 X 51 = 2,550Thus the sum of al l even numbers from 1 to 100 is 2,550.In Short Cut 2 th e su m of al l the numbers from 1 to 99 isfound to be 4, 950; consequently the sum of al l numbers from1 to 100 is 5,050. In Short Cut 3 th e su m of al l odd numbersfrom 1 to 100 is found to be 2,500. Our answer fo r th e su mof al l the even numbers from 1 to 100 is therefore in agreement.

Sum ofall numbers

5,050Su m of

all odd numbers2,500

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Sum ofal l even numbers2.550

5ADDING A SERIES OF NUMBERS WITH ACOMMON DIFFERENCE

Sometimes i t is necessary to add a group of numbersthat have a common difference. No matter what the common difference is an d no matter how many numbers ar e be ing added, only one addition, multiplication, and divisionwill be necessary to obtain the answer.

Rule: Add the smallest number to the largest number,multiply the su m by the amount of numbers inth e group, and divide by 2.

As an example, le t us find the su m of the followingnumbers:

87, 91, 95, 99, an d 103Notice that the difference between adjacent numbers isalways 4. This short-cut method can therefore be used.Add th e smallest number, 87, to the largest number, 103.Multiply the sum, 190, by 5, since there ar e five numbersin the group.

190 x 5 = 950 (Short Cut 11)Divide by 2 to obtain the answer.

950 + 2 = 475 AnswerThus 87 + 91 + 95 + 99 + 103 = 475.

(Naturally, this is exactly the same as the rule in ShortCut 1, because there we were simply adding a series ofnumbers with a common difference of one. So, for ease ofremembering, you can combine Short Cuts 1 an d 5.)

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6ADDING A SERIES OF NUMBERS HAVING ACOMMON RATIO

Rule: Multiply th e ratio by itself as many t imes asthere ar e numbers in th e series. Subtract 1from th e product and multiply by th e firstnumber in th e series. Divide th e result byone less than the ratio.

This rule is best applied when th e common ratio is asmall number or when there ar e few numbers in th e series.If there ar e many numbers and the ratio is large, th e ne-cessity of multiplying th e ratio by itself many times di-minishes the ease with which this short cu t can be applied.But suppose we ar e given th e series:

53, 106, 212, 424Here each term is twice the preceding term, and there ar efour terms in the series. Th e tatio, 2, is therefore multi-plied four times.

2 X 2 X 2 X 2 = 16Subtract 1 and multiply by th e first number.

16 - 1 = 15; 15 x 53 = 795 (Short Cut 27)Th e next step is to divide by one less than the ratio; how-ever, since th e ratio is 2, we need divide only by 1.Thus the sum of ou r series is

53 + 106 + 212 + 424 795 Answer

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SHORT CUTS IN ADDITION 15Practice Exercises fo r Short Cuts 1 through 6

Find the su m in each case.

1) All odd numbers from 1 to 23 =

2} 3. 6 12 + 24 + 48 + 96

3) All numbers from 84 to 105

4} 56 + 59 + 62 + 65 =

5) 24 + 72 + 216 =

6) 14+15+16+17+18+19+20+21 =

7) All numbers from 1 to 1,000 =

8) All even numbers from 1 to 50

9) 132 + 137 + 142 + 147

10) 197 + 198 + 199 + 200 + 201 + 202 + 203 =

. . . . . . . . . . . . . - . . . . . . . , - - - - - - - - - - - - - - - - - - - - - - -


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Chapter 2SHORT CUTS IN MULTIPLICATION

Multiplicatwn is itself a short-cut process. Fo r example, a problem in repeated addition,

3 + 3 + 3 + 3 + 3 + 3 + 3 = 21is quickly recogmzed as nothing more than

7 X 3 = 21This shorthand notation led us directly to the answer,eliminating the necessity of six additions along the way.

Fo r most of us, the multiplication table, drummed intoour minds early in our mathematical training, provided thereference source fo r obtaining the answer. But, happily,proficiency in multiplication does not depend on memorizing tables. The short-cut methods described in this section employ addition, subtraction, division, and, of course,elementary multiplication. But i f you can add two numbersquickly and halve or double a number w1th ease. you shouldhave no trouble at all.

THE DIGITS

The basic calculating unit is the digit. When two numbers are multiplied, every combination of their individualdigits is multiplied, and by correctly adding the results(with proper r egard to their position) the product of thetwo numbers is obtained.

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18 SHORT CUTS IN MULTIPLICATIONConsider the following example:

432 X 678The nine possible combinations of dig1ts of the two numbersar e

4 X 6;4 A 7;4 X 8;

3 X 6;3 X 7;3 X 8;

2 A 62 X 72 X 8

By arranging the products according to number position,we can obtam the product desired.2 4 1 8 1 2 2, 7 1 2

2 8 2 1 1 4 2,0 3 43 2 2 4 1 6 1,3 56

2,7 1 2 2,0 3 4 1,3 5 6 292 ,896

432 X 678 2 9 2, 8 9 6 Answer

Thus, by memorizing only the multiplication tables for al ldigits from 1 to 9 we ar e able to multiply one number byanother, regardless of how many digits each of them contains.

But memorizing the eighty-one products in the multlplication table is not essential for multiplying by the digits.The methods for multiplying by the d1gits described in thissection involve only addition, subtractwn, and doubling orhalving.

The rules ar e given in detail intentionally. Fo r somedigits, the rule may appear unusually long. This is onlybecause the presentation must consider all exigencies.Don't be discouraged by what seems like a complicated

SHORT CUTS IN MULTIPLICATION 19way of multiplying a simple digit. After the second or thirdread ing of the rule a pattern will emerge and the processwill become a mere routine.A rule for multiplication by 1 has been omitted, since th eproduct obtained by multiplying any number by 1 is the original number.


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7MULTIPLYING BY NUMBERS ENDING IN ZEROS

Numbers ending in zeros may be thought of as th e prod-uct of the nonzero part multiplied by a power of 10 Foexample, 37,000is really 37 x 1,000. Since m u l t i ~ l y i n ;by zero results 1n zero, multiplying by numbers ending in

z . e ~ o s may be shortened by ignoring th e zeros and then af -f v o n ~ required amount after th e nonzero part has beenmultlphed.

Rule: ~ u l t i p l y th e two numbers as i f they did not endm zeros . Then affix an amount of zeros equalt? s.um of al l th e zeros ignored in th e mul-tlphcatlOn.

A simple case will be chosen. Let us find the product of37,000 X 6,000,000

By ignoring th e zeros, we have37 X 6

Using S h o r ~ Cut 12, we flnd 37 x 6 = 222. A total of ninezeros was 1 g ~ o r e d before the multiplication; therefore ninezeros ar e aff1xed to th e product.

222, 000, 000, 000 Answer

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8MULTIPLYING BY 2

Multiplying by 2 is another way of saying we are doubl-ing a number or simply that we ar e adding a number to it -self. Doubling a number may be accomplished quicklywithout carrying by applying th e following simple rule.

Rule: Starting from th e first digit of th e given num-ber, double the digit if 1t is 4 or less and putth e answer under the respective digits of th egiven number. Fo r digits 5 to 9, subtract 5and double th e result. Place the answer un -de r th e respective digits of the given num-ber. Now inspect the tentative answer. Eachdigit of th e answer to the immed1ate left ofa digit in th e given number 5 or greatershould be increased by 1. Th e result is th efinal answer.

At first reading, this rule ma y sound more complicatedthan simply add1ng digit by dig1t. Th e beauty of th1s short-cut method is , however, that the answer is obtained im -mediately from left to right and we ar e never bothered byhaving to remember to carry over any digits. As an ex -ample, suppose we were asked to multiply 5,377 by 2. Firstlet us write th e given number, using ou r alphabetic identi-fication:

A B C D5 3 7 7

Starting from A, double each number less than 5 (but notequal to 5); if the number is greater than 5, subtract 5from it and double th e result, placing a small line undereach digit of th e answer that is to the immediate left of a

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22 SHORT CUTS IN MULTIPLICATIONdigit in the given number that is 5 or more. The reasonfor this small line will be explained shortly. In our givennumber, the first digit is 5; subtract 5 from this and doublethe result.

5 - 5 = 0; 0 + 0 = 0Place 0 under the 5 and a small line under the space to theleft of the 0 (since there is no number in that space). Ourfirst result will look like this:

A B C D5 3 7 70

Given numberTentative answer afterfirst step

The next digit is less than 5, so we merely double it, andour answer begins to look like this now:

A B C D5 3 7 70 6

Given numberTentative answer after

second stepThe C digit is a 7; subtract 5 from this and double theresult.

7 - 5 = 2; 2 + 2 = 4This is the C digit of the answer; but remember, a smallline must be placed under the next digit to the left in theanswer (the 6). We have now come this fa r in ou r answer:

A B c D5 3 7 7 Given number0 6 4 Tentative answer afterthird step

Finally, the D digit is more than 5, so once again we ob-tain 4 and place a small line under the previous 4 in theanswer. Our answer now looks like this:

T CUTS IN MULTIPLICATIONSHOR23- A B C D

5 3 7 70 6 4 4

Given numberTentative answer afterfourth step

d L. ed digit is increased by 1 to obtain the finalEach un er lnanswer. 1o, 754 Answer


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9MULTIPLYING BY 3

Rule: Th e first tentative digit of th e answer is ob tained by taking one-half the first digit of thegiven number.

Next, in turn, each digit of th e given numbe r is subtracted from 9, th e result doubled,then added to one-half th e digit to th e rightin the given number to obtain each digit ofth e answer. I f th e original digit in the givennumber is odd, add an extra 5. Ignore anyfraction that occurs when taking one-half anumber.

To find the units digit of the answer, subtract th e units digit of th e given number from1 0 and double th e result. Add an extra 5 ifth e units digit of th e given number is odd.In each of th e steps above, record only th eunits digit in the answer. Any tens digitshould be carried and added to th e answerdigit immediately to th e left.

To obtain th e final answer from the tentative answer digits obtained above, subtract2 from the first digit recorded.Naturally, when multiplying a small number by 3, th e

"long" way would probably be as quick, though maybe notas simple to use; but when long numbers ar e multiplied,th e short cut explained above is an excellent time and laborsaver.

Fo r example: 4,635,117 x 3.One-half of 4 is th e first tentative digit.

4) = 224

SHORT CUTS IN MULTIPLICATION 25di 't of the answer is found by subtracting 4 fromThe next gl lt and adding one-half the digit to th e9 d ubling th e resu ' dd d0 s ce 4 is even, th e additional 5 is not a e right, 6. m4 _ 5 5 X 2 = 10; 10 + ~ ( 6 ) = 139 - - '

d th 3 carry th e 1 and add it to th e 2 previouslyRecor e , 'determined. . 6Th e next digit in the given number lS

9 - 6 = 3; 3 X 2 = 6; 6 + H3) = 7(Th e fraction ignored. ) Th e next digit in th e givennumber is 3.

9 - 3 = 6; 6 X 2 12; 12 + i(5) = 1414+5 = 19

(The 5 was added because 3 is odd.) Record 9;. carrythe 1 to the left. The four digits thus far obtamed ln th eanswer ar e3 3 8 9Continue with th e other digits of th e given number.

9 - 5 = : 4 X 2 = 8; 8 + i( 1) = 88 + 5 = 13

Record th e 3; carry th e 1 to the left.9 - 1 = 8; 8 X 2 = 16; 16 + t


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26 SHORT CUTS IN MULTIPLICATIONRecord 1; carry 1. The digits obtained ar e

33,905,351The final step involves subtracting 2 from the first digit.

13,905,351 Answer

10MULTIPLYING BY 4

Rule: The first tentative digit of the answer will beone-half the first digit of the given number.Ignore any fractiun in this and other steps. Th eother answer digits are found by subtractingeach of the digits of the given number from 9and adding one-half th e digit to the right. Ifthe digit of th e given number is odd, add anextra 5.To find the units digit of the answer, subtract the units digit of the given number from10 . Add 5 i f the units digit of the given numbe r is odd. To obtain the final answer, subtract 1 from the first digit recorded.In each of th e cases, above, if the result ofone of the steps is a two-digit number, re cord th e units digit an d carry any tens digitleft to the preceding answer digit.

As an example: Multiply 37,485,109 by 4.The first tentative digit is one - half the first digit of thegiven number, 3.

~ ( 3 ) = 1(Ignore the fraction.)

In each of th e next steps, subtract the digit of the givennumber from 9, add one-half the digit to the right, and add5 more i f the digit in the given number is odd.9 - 3 = 6; 6 + ~ ( 7 ) + 5 = 14

(Her e 5 is added because 3 is odd.)Record the 4 an d add 1 to the 1 previously determined.27


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28 SHORT CUTS IN MULTIPLICATIONTh e next digit in the given number is 7.

9 - 7 = 2; 2 + ~ ( 4 ) + 5 = 9(Agam, 5 is added because 7 is odd.) Continue in turn with4, 8, 5, 1, and 0.

9 - 4 5; 5 + ~ ( 8 ) 99 8 = 1' 1 + t{5) 39 - 5 = 4' 4 + t{1) + 5 99 - 1 = 8' 8 + ~ ( 0 ) + 5 13Record 3; carry 1 to the left.9 - 0 = 9; 9 + t(9) =13

(The zero is considered even.) Record 3; carry 1 forward.We have now reached th e units digit of th e given number.To obtain the units digit of the answer, subtract the unitsdigit of the given number from 10. Add 5, since it is odd.

10 - 9 = 1; 1 + 5 = 6We have now obtained th e following tentative answer:

249,940,436Th e final answer is obtained by subtracting 1 from the firstdigit, 2.

149,940,436 Answer

11

MULTIPLYING BY 5When any digit is multiplied by 5, the units digit of theoduct is always either 5 or 0 an d the tens digit is alwaysprual to one-half the given digit (ignoring the fraction t>.

interesting property of 5 leads us to the first of twoshort-cut methods for multiplying by 5.First Method

Rule: Th e first digit of the answer is equal to onehalf the first digit of the given number. Eachsucceeding answer digit is equal to 5, i f thecorresponding digit in th e given number isodd; or 0, i f the corresponding digit in thegiven number is even; plus one-half of thedigit to the right in the given number. Theunits digit of the answer is 5, if the given number is odd; an d 0, if the given number is even.Ignore any fraction resulting from the halvingprocess.

Second MethodRule: Move the decimal point of th e given number

one place to the right and divide the resultingnumber by 2.Although th e second method seems simpler at first

reading, both methods are equally easy to employ an d bothwill find applications, depending on the problem. Usuallyfor small even numbers, the second method would probablybe used more often. However, both methods will be demonstrated, using the same given number.

29


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-30 SHORT CUTS IN MULTIPLICATIONMultiply 78,439 by 5. 12

A B C D E F7 8 4 3 9 Given number

F ~ r s t Method. The first digit of the product (the A digit)w1ll be equal to one-half of 7 (ignoring theA3 First digit of product

Since the B digit of the given number is odd, the B digit ofthe product will be 5 plus one-half the C digit of the givennumber (5 + 4 = 9). The C digit of the given number iseven, so tha.t C digit of the product will be 0 plus onehalf the D d1g1t (0 + 2 = 2) . Th e D digit of the product is0 + 1 = 1. TheE digit of the product is 5 + 4 = 9. TheF. digit of the product is the units digit in this case, ands1nce the units digit of th e given number is odd th e unitsdigit of the product will be 5. The final product is

ABCDEF3 9 2, 1 9 5 Answer

Seco nd Method. Move the decimal point of the given numbe r one place to the right.7 8, 4 3 9 . 0 becomes 7 8 4, 3 9 0.

Divide the new number by 2.7 8 4, 3 9 0 + 2 = 3 9 2, 1 9 5 Answer

Th e same result was obtained with the first method.

MULTIPLYING BY 6Rule: The first digit of the answer is one-half th e

first digit of the given number.Th e other answer digits are obtained byadding each of the digits of the given numbe r to one-half the digit to it s right. Anextra 5 is added i f the given digit is odd.

Ignore any fraction that occurs when halving a number .The units digit of the answer is the unitsdigit of the given number, if even. I f odd,add 5 to the units digit of the given numberto obtain th e units digit of the answer.In each case, i f the result is a two-digitnumber, record only the units digit. Carryany tens digit left and add it to the preceding answer digit.

This short cut may seem like a roundabout way to multiply by 6, but the opposite is actually true. In fact, the .beauty of this method is the simplicity and ease with wh1chan answer may be written directly. You will soon findyourself able to multiply any number by 6, using only alittle quick mental addition without bothering to write anyintermediate steps.As an example of the procedure, we shall multiply

714,098 X 6The first digit (tentatively) will be one-half of 7, or 3(neglecting the t, of course). The answer digits that followwill depend on whether the corresponding digits in th egiven number are odd or even. Since the first digit is odd,

31


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32 SHORT CUTS IN MULTIPLICATIONadd 5 and one-half the next digit to the right.

7 + 5 + ~ ( 1 ) = 7 + 5 + 0 = 12(Remember, ignored.) Record the units digit 2 andcarry tens digit t.o the left to be addeci to the 3 previously wntten. Th e f1rst two answer digits ar e

4 2Th e next digit of the given number is 1, which is also odd.

1 + 5 + i(4) = 1 + 5 + 2 = 8Record th.e 8 ~ o v e on to the next digit in the given number, 4. Smce this 1s even, merely add to it one-half thenext digit to the right in the g1ven number.

4 + HO) = 4Re cord this in the answer. Thus fa r we have determinedthe following d1gits in the answer:

4 2 8 4Th e next digit is 0 (which is considere d even). Therefore,add one-half the next digit to th e right.

0 + i{9) = 4 (ignoringTh e next d1git is 9, which is odd.

9 + 5 + ~ ( 8 ) = 9 + 5 + 4 = 18Record the 8 in th e answer and carry the tens digit, 1, left

be added to the preceding digit, 4. The units digit of theg ~ v e n number is next; since it is even, it is also the unitsd1git of the answer.

The product is therefore4, 2 8 4, 5 8 8 Answer

13MULTIPLYING BY 7

Rule: The first tentative digit of the answer is onehalf the first digit of the given number.Th e rest of the answer digits ar e obtainedby doubling the digit of the given number and

adding one-half the digit to it s right. Add anextra 5 if the given digit is odd . Th e unitsdigit of the answer is twice the given unitsdigit. Add 5 i f the given units digit is odd.Ignore any fraction that may occur. Recordonly the units digit in each case. Any tensdigit should be carried and added to theanswer digit immediately to the left.

Example: 97,841 x 7.The first digit is one-half 9.

~ ( 9 ) = 4(Ignore the fraction.) Next, in turn, double each digit ofthe given number, add one-half the digit to the right, andadd an extra 5 if the given digit is odd.

9 X 2 = 18; 18 + ~ ( 7 ) = 21 21 + 5 = 26Record 6 and add 2 to the preceding answer digit, 4.

7 X 2 = 14; 14 + i(8) = 18 18 + 5 = 23Record 3; carry 2 to the left.

8 X 2 = 16; 16 + %(4) 18Record 8; carry 1.

4 X 2 = 8' 8 + ~ ( 1 ) 833


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34 SHORT CUTS IN MULTIPLICATIONFinally, the units digit of the answer is determined.

1 X 2 = 2; 2 + 5 = 7The digits obtained are

6 8 4, 8 8 7 Answer

14MULTIPLYIKG BY 8

Rule: Write the first digit of the given number as thefirst tentative digit of the answer. The nextanswer digit is obtained by subtracting the firstdigit of the given number from 9, doubling theresult, and adding the second digit of the givennumber. Continue the process by subtractingeach digit of the given number from 9, doublingthe result, and adding the next digit to its right.To obtain the units digit of the answer, simplysubtract the units digit of the given numberfrom 10 and double the result. In each of thesteps above, record only the units digit of thesum; any tens digit should be ca rried and addedto the preceding answer digit. To obtain thefinal answer, subtract 2 from the first digitobtained.

A typical example is sufficient to show how this shortcut works.Example: 379,146 x 8.First, write the 3 as the tentative first digit of theanswer. Next subtract 3 from 9, double the result, andadd the next digit to it s right, 7.

9 - 3 = 6; 6 X 2 = 12; 12 + 7 =19Record the 9, carry the 1, and add it to the 3 previouslyrecorded. The first two tentative digits of the answer ar e

4 9Proceed with the next digit, 7.9 - 7 = 2; 2 X 2 = 4; 4 + 9 = 13

35


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36 SHORT CUTS IN MULTIPLICATIONRecord the 3, carry the 1, and add it to the previously de termined 9. But 9 + 1 = 10. Therefore record the 0 andcarry the 1 another digit to the left, adding it to th e 4. Thefirst three digits ar e now

50 3Continue ttus procedure .

9 - 9 = 0;Record 1.

9 - 1 = 8;

0 X 2 O'8 X 2 = 16;

0 + 1 = 1

16 + 4 = 20Record 0 and add the 2 to the 1 preceding. Th e answerdigits obtained to this point are

50 3 3 0Continue with the process.

9 - 4 = 5; 5 X 2 = 10; 10 + 6 = 16Record the 6; carry the 1. Th e next digit is the units digitof the given number. Subtract this from 10 and double theresult. Record th e result as the units digit of th e answer.

10 - 6 = 4; 4 X 2 = 8The tentative answer to the problem is

5, 0 3 3, 1 6 8To obtain the final answer, we must subtract ? from th efirst digit, 5.

3, 0 3 3, 1 6 8 Answer

15MULTIPLYING BY 9

Rule: The first digit of the given number minus 1 isthe first digit of the answer. Th e second digitof the answer is obtained by subtracting the firstdigit of the given number from 9 and adding_ itto the second digit of the given number. Contmuethis process by subtracting each digit in the given number from 9 and adding to the result thenext digit to it s right. Stop this procedure afterthe tens digit of the answer is obtained. Th e unitsdigit of the answer is obtained by subtracting theunits digit of the given number from 10. Ineach case if the sum is a two-digit number,record units digit and carry the tens digitto the preceding answer digit.

Multiply 7,149 by 9. . . .The first digit of th e given number mmus 1 1s the flrstdigit of the answer.

7 - 1 = 6The second digit of th e answer is 9 minus th e f i r ~ t digitof the given number plus the second digit of the g1vennumber .

9 - 7 = 2; 2 + 1 = 3We now have th e first two dig1ts of the answer (a t leasttentatively) .

6 3To obtain the third digit of the answer, subtract the seconddigit of the given number from 9 and add the result to thethird digit of th e given number.

9 - 1 = 8; 8 + 4 = 1237


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38 SHORT CUTS IN MULTIPLICATIONHere the result is a two-digit number. Th e units digit isrecorded as part of the answer, and the tens digit is carried and added to the 3 previously determined. The firstthree digits of th e answer are now

6 4 2The tens digit and th e units digit of th e given number areused to obtain the tens digit of the answer.

9 - 4 = 5; 5 + 9 = 14Record the 4, ca rr y th e 1, and add it to th e 2 previously determined. The units digit of the answer is merely 10 minusthe units d1git of th e given number.

10 - 9 = 1Th e product is therefore

7,149 X 9 64, 341 Answer

SHORT CUTS IN MULTIPLICATION 39Practice Exercises fo r Short Cuts 7 through 15

1) 47,821 X 5 =2) 8,300 X 2,000,000 =3) 7,914 X 8 =4) 682 X 9 =5) 1,356 X 7 =6) 51,007 X 2 =7) 6,045 X 6 =8) 497 X 3 =9) 12,760,195 X 4 =

10) 1,116 X 911 ) 436 X 5 =12) 31.875 X 313) 613,767 X 7 =14) 44,060 X 6 =15) 831,615 X 8 =


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NUMBERS BEGINNING OR ENDING IN 1

. When a given number is multiplied by 1, the product1s the same given number. This unique property of 1 isused to good advantage in numerous short cuts. When a~ u l t i p l i e r containing 1 is used, somewhere in the answer1s the number bemg multiplied. This fact forms the basisof many of the short cuts that follow.

40

16MULTIPLYING BY 11

Rule: The first digit of th e given number is the firstdigit of th e answer. Add the first digit to thesecond digit of the given number to obtain thesecond digit of the answer. Next, add thesecond digit of th e given number to the thirddigit of the given number to obtain the thirddigit of the answer. Continue adding adjacentdigits until th e tens digit of the given numberis added to the units digit of the given numberto obtain the tens digit of the answer. Theunits digit of the answer will be the units digitof the given number. If any of the sums aretwo-digit numbers, record only the units digitand add the tens digit to the preceding answerdigit.

Two examples will best show how to use this short cut.Example No. 1: Multiply 81,263 by 11.

Th e first digit of th e answer will be 8, the first digitof the given number. The second digit of the answer willbe the sum of the first and second digits of the given number , 8 + 1 = 9. Continuing from left to right, the sum ofadjacent digits in th e given number will produce digits ofthe answer. Th e result is shown below:Given numberAnswer

41


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42 SHORT CUTS IN MULTIPLICATIONIn the example above, each su m was less than 10. But whatwould happen if the sum was 10 or more?

Example No. 2: Multiply 67,295 by 11.Th e 6 is the tentative first digit. Th e second dig1t is the

su m of 6 and 7, or 13. Here the sum is greater than 10.Th e 3 becomes the tentative second digit of th e answer, butthe 1 is carried left and added to the first digit.

6 + 1 = 7This IS the new first digit of the answer. The third digit ofthe answer is found by adding th e second digit of the givennumber to the third digit, 7 + 2 = 9. Th e next digit in theanswer is 2 + 9 = 11. Again the units digit becomes partof the answer, and the tens digit is carried left to the previously determined answer digit.

9 + 1 = 10.The 0 is the new third digit of the answer, and the 1 iscarried still further left to the second digit, 3 + 1 = 4.This is the new second digit. Continue in this fashion untilal l adjacent digits have been added. Th e final digit in theanswer is 5. This process is shown pictorially thus:

17

MULTIPLYING BY 12Rule: Precede the given number with a zero. Start

ing from this zero, double each digit and addto it the next digit to it s right. Record th esum. When the units digit of the given numberis reached, simply double it and record th esum. In each step, if the doubling processresults in a two-digit number, record only th eunits digit and add the tens digit to th e preceding answer digit.

This simple short cut is particularly handy when wewant to project some monthly event over the entire year.Suppose we are asked to find th e total rent paid. during theyear i f the monthly rental is $132.50. To do th1s we mu l-ti ply the monthly rental by 12. Our problem then becomes

$132.50 X 12First, place a zero in front of the number.

0 1 3 2. 5 0Next double each digit and add to it the digit to th e right.1 to 0 gives the first digit of the answer, 1. Addingtwice 1 (the second digit of the given number) to 3 (itsneighbor to the right) gives th e second digit of the answer,5. Continuing in this manner. we obtain th e answer.

0 1 3 2. 5 0 Given number$ 1, 59 0. 0 0 Answer

Notice that in doubling th e 5 in the given number the re sult was 10. Th e 0 was recorded and the 1 was added to th epre ceding digit. The precedin g digit, however, was a 9,which when increased by 1 became 10. Again the 0 was43


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44 SHORT CUTS IN MULTIPLICATIONr e ~ o r ~ e d a.nd. th e 1 again carried another step to th e left.

T h l ~ trme 1t tncreased the previously determined 8 to a 9.Until th e 5 was doubled, th e answer digits were 1,589. 18

MULTIPLYING BY 111Rule: Imagine a number whose digits are

ABCDEFGH I JKLMTh e first digit of the answer will be A. Th esecond digit will be A + B. The third digit willbe A+ B + C. Th e fourth digit will be B + C + D.Th e fifth digit will be C + D + E. This procedureis followed, adding three adjacent digits to gether, until th e final three digits are reached.The hundreds digit of th e answer will be K +L + M. The tens digit of the answer will beL + M. The units digit of the answer will alwaysbe the units digit of th e given number, in thiscase, M. Remember that whenever th e su m isa two-digit number, the units digit is th e answerportion and th e tens digit is added to th e preceding answer digit. Thus, if I + J + K is a twodigit number, the tens digit will be added to thesu m of H + I + J previously determined.

Follow th e next example step by step.6 5 9, 8 4 5 X 1 1 1

Th e first digit of the answer will be the first digit of thegiven number, 6. The second digit of the answer is6 + 5= 1 1

Write the 1 and carry th e tens digit (also 1) left.6 + 1 = 7The next digtt is6 + 5 + 9 = 20

45


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46 SHORT CUTS IN MULTIPLICATIONWrite the 0 and carry the 2 left.

1 + 2 = 3Th e three digits we have found thus fa r ar e730Now begin adding the digits of the given number in groups

of three .5 + 9 + 8 = 22Write 2; carry 2.9 + 8 + 4 = 21Write 1; ~ a r r y 2. Continue this process until the lastthree dlglts, 845, ar e reached.

8 + 4 + 5 = 17Write 7; carry 1.

4 + 5 = 9Write 9; no carry.

final digit of the answer will be the units digit ofthe g1ven number, 5.In pictorial form, the entire example looks like this:

4+5 5l l9 5

I 573,242,795 Answer

19MULTIPLYING BY A MULTIPLE OF 11

Rule: Multiply by the units digit of the multiple of 11(using the appropriate short cut). Then multi-ply by 11 (Short Cut 16).

Although the beginner will usually apply this short cutin two separate operations, as he becomes more expert init s use the final answer will be obtained in only one opera-tion. The explanation given below is in two distinct steps,since this presentation is easier to follow.

Multiply 84,756 by 66 .Here we are multiplying by the sixth multiple of 11,since 6 x 11 =66. First, apply Short Cut 12 for multiplyingby 6. Next, multiply the result by 11, using Short Cut 16.

6 X 84,756 = 508,53611 X 508,536 = 5,593,896

Therefore84, 756 x 66 = 5, 593, 896 Answer

47

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49


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20MULTIPLYING BY 21

Rule: Th e first di git (or digit s ) of the answer will betwi ce the first digit of the given number. Thes econd di git of the answer will be th e firstdi git of th e given number plus twice th e se conddigit of the given number. The third digit ofth e answe r will be the second digit of the givennumber plus twi ce the third digit of the givennumber. Continue the process until the tensdigit of the given number is added to twice theunits digit of th e given number. This su m isthe tens digit of the answer. The units digitof the answer is th e units digit of the givennumbe r. Whenever a su m is a two-digit number, reco rd it s units digit and add th e tensdigi t to the pr eceding answer digit.

This ru le is ve ry muc h like the one for multiplying by11. In fac t, s ince 21 is the su m of 11 and 10, it does belong to the sam e family of short cuts .

As an example, we shall multiply 5,392 by 21.The fi rs t d1git s of the answer will be equal to twic e the

first d1git of the given number.5 X 2 = 10

Next, add th e first di git of th e given number , 5, to twicethe se cond digit , 3.

5 + (2 X 3} = 11Th e units digit becomes the next answer digit, and the tensdigit is added to the 10 previously determined. Th e firstthree digits up to this point are

1 1 148

sHORT CUTS IN MULTIPLICATION-The next digit is obtained by adding 3 t o twice 9.3 + (2 X 9) = 21

Record the 1 and carry the 2 to the left. The first fourdi gits of the answer are now

1 1 3 1The tens digit of the answer is obtained a ~ d ~ n g the tensdigit of the given number to twice th e umts d1g1t of th egiven number.

9 + (2 X 2) = 13Record the 3; carry th e 1 to the left. The units di git of theanswer is the units digit of th e given number , 2.

Th e product is therefore5 , 392 x 21 = 113, 232 Answer


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21MULTIPLYING BY 121

Rule: Multiply th e given number by 11, using ShortCut 16. Multiply the product obtained by 11again.

Th e ease with which Short Cut 16 can be used permitseven a two-step method such as this to be applied withrapidity. When used with small numbers, say, two- orthree-digit numbers, the numbers obtained in th e firststep ma y be retained in the mind and th e second stepperformed by writing th e answer immediately . In th esample problem, th e two steps will be shown.

Multiply 8,591 by 121.Multiply the given number by 11, using Short Cut 16.

8,591 X 11 = 94,501Multiply th e result by 11.

94,501 x 11 = 1,039,511 Answer


Rule: First, write th e first two digits of the givennumber as th e first two answer digits. Then,starting from the third digit of th e given number, add each of th e digits of th e given numbe r in turn, adding th e third digit to the firstdigit, the fourth digit to the second digit, andso on. When th e last digit of th e originalnumber is reached, continue writing the remaining digits of th e given number.

Fo r example:164,759 X 101.

Th e first two answer digits ar e1 6

Starting from the third digit, 4, add in turn th e digits ofth e given number, 1-6-4-7-5-9.

1 + 4 :: 5; 6 + 7 = 13; 4 + 5 = 9; 7 + 9 = 16Th e 9 is th e last digit of th e original given number. Thereafter merely record the balance of the digits of the givennumber not added: in this case, 5 and 9. Naturally, in th eadditions performed above, th e units digit is recorded asth e answer digit; any tens digit is added to the precedinganswer digit.Therefore

164,759 x 101 = 16,640,659 Answer

50 51


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23MULTIPLYING BY 1,001

Rule: First, write the first three digits of th e givennumb_er as th e first three answer digits. Then,startmg from the fourth digit of the given num-ber, ad d each of the digits of the given numberin t u ~ n , a ~ d ~ n g the fourth digit to the first digit,the fifth d1g1t to the second digit, an d so on.When the last digit of the original given num-b ~ r . is reached, continue writing the remainingd1g1ts of the given number.

Fo r example: 23,107 x 1,001.The first three answer digits are

2 3 1S t a r t ~ n ~ from the _fourth digit of the given number, o, addthe d1g1ts of the g1ven number in turn.

2 + 0 = 2; 3 + 7 = 10The 7 1s the last digit of the original given number there-fore the digits of the given number not ye t added, 1', o, and7, are merely written as the answer digits.

. ~ h e n e v e r a su m is greater than 9, record th e unitsd1g1t and add the tens digit to the preceding answer digit.23,107 x 1, 001 = 23,130,107 Answer

52

24MULTIPLYING BY ONE MORE THANA POWER OF 10

Rule: Write as many digits of the given number asthere are digits in the multiplier less one.Then, starting from the digit whose place isequal to the number of digits in th e multiplier,add, digit by digit, the given number to theoriginal given number.

What this rule means is that i f th e multiplier has sevendigits, the addition should start from the seventh digit. Thefirst digit of th e given number is to be added to the seventhdigit of th e given number, the second digit added to theeighth, and so on. Naturally, the first si x digits of th eanswer will be the same as the first si x digits of the givennumber unless they are changed by some digit that iscarried forward.For example: 66,809,542 x 100,001.There are si x digits in the multiplier; therefore writethe first five digits of the given number as the first fiveanswer digits. Starting at the sixth digit of th e given num-ber, add the digits of the given number.

6 + 5 = 11; 6 + 4 = 10; 8 + 2 = 10The balance of th e digits not added are merely written asgiven in th e original number. When the sums are two-digit numbers, record the units digit as part of the answerand add the tens digit to the preceding answer digit. Thus,in the three sums shown above, 1 is recorded and 1 is car-ried forward; 0 is recorded an d 1 is carried forward; 0 isrecorded and 1 is carried forward. Th e rest of th e digitsar e recorded as they appear in the original given number.

0 9 5 4 253


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54 SHORT CUTS IN MULTIPLICATIONThe product is therefore

66,809,542 x 100,001 = 6,681,021,009,542 AnswerWhich, in case you ar e interested, can be read as : Sixtrillion; si x hundred eighty-one billion; twenty-one million;nine thousand; five hundred and forty-two.

25MULTIPLYING "TEEN" NUMBERS

Rule: To one of the numbers, add the units digit ofthe other number. To the result, affix the unitsdigit of the product obtained by multiplying theunits digits of th e given numbers. Any tensdigit should be added to the sum found in thefirst step.

The teen numbers include al l numbers from 10 to 19.Example: 13 x 17.The units digit of the first number may be added to the

second number, or the units digit of the second numbermay be added to the first number. In either case the resultis the same.

13 + 7 = 20 or 17 + 3 = 20Affix the units digit of th e product obtained by multiplyingthe units digits of the given number.

7 X 3 = 21Affix the units digit, 1; the tens digit, 2, 1s added to thesum found in th e first step.

20 + 2 = 22Thus

13 x 17 = 221 Answer

55


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26MULTIPLYING BY At-."'Y TWO-DIGIT ~ U M B E RENDING IN l

Rule: Multiply the first digit of the given number bythe tens digit of the multiplier. The productis the first digit (or digits) of the answer. Thenext digit is obtained by adding the first digitof the given number to th e product of the seconddigit of the given number and the tens digit ofthe multiplier. Continue this process until thetens digit of the given number is added to theproduct of the units digit of the given numberand the tens digit of th e multiplier. This willbe the tens digit of the answer . Th e units digitof the answer will always be the units digit ofthe given number. Notice that only the tensdigit of the multiplier is used in the varioussteps. Keep in mind that whenever a two-d1g1tsu m 1s obtained, the units digit is recordedwhile the tens digit is added to the precectinganswer dig1t.

Th e beauty of these general short cuts is that they per-mi t the ch01ce of many different methods for obtaining ananswer, depending on the tens digit. I f we were called uponto multiply by 91, the rule above might not be as easy touse as some other rule. fo r example. Short Cu t 60.

Multiply: 843 X 31.The first digits of the answer will be three times the

first digit of the given number .8 X 3 : 24

The next digit is the sum of the first digit of the given num-ber, 8, and three times the second digit of the given number,

56

sHORT CUTS IN MULTIPLICATION 57

4. The three, of course, comes from the tens digit of themultiplier, 31.8 + (3 X 4) = 20

R cord the 0 and carry th e 2 to the left. Next, add 4 toth:ee times 3 to obtain the tens digit of the answer.

4 + (3 X 3) = 13Record the 3 an d carry th e 1. So far, ou r answer looks likethis:

2 6 1 3Only the units digit is ye t to be d e t e r m i n ~ d . . .The units digit of the answer is the umts d1g1t of thegiven number, 3.

Therefore843 x 31 = 26,133 Answer


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58 SHORT CUTS IN MULTIPLICATIONPractice Exercises fo r Sfwrt Cuts 16 through 26

1) 6,528 X 33 =2) 172,645 X 113) 956 X 1214) 13 X 18 =5) 2, 742 X 1,001 =6) 24 , 863 X 217) 726 X 111 =8) 2,665 X 12 =9) 547 X 10, 001 =

10) 42 X 111 =11 ) 23,316 X 11 =12) 167 X 101 =13) 74 , 155 X 4114) 89 X 12 =15) 1,038 X 121

NUMBERS BEGINNING OR ENDING IN S

The number 5 is perhaps the mos t interesting one towork with as well as one of the simplest. When we multi-ply a number ending in 5 by any other number, th e unitsdigit of the product iS always either 0 or 5, depending onwhether the given number is even or odd. In fact, the easewith which 5 is multiplied permits us to adapt short c uts tonumbers having 5 in a position other than at either end.Short Cut 36, fo r example , can be applied even when 5 ap -pears in the middle of a number. Thus, although this sec-tion concerns itself particularly with numbers having 5 ateither end, the methods are by no means restricted tosuch numbers.

59

SHORT CUTS IN MULTIPLICATION 61


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27MULTIPLYING BY 15

Rule: Add one-half th e first digit to itself to obtainthe first answer digit (or digits).

Continue this process until the units digit isreached. Add an extra 5 i f the digit to the leftof the given digit is odd.

If any of the sums is more than 9, recordthe units digit and add the tens digit to thepreceding answer digit.

Ignore any fractions that may occur.If the units digit of the given number is even,

the units digit of the answer is 0. I f the unitsdigit of the given number is odd, the units digitof the answer will be 5.

Fo r instance, multiply 738 by 15.Add 7 to one-half itself, ignoring th e fraction.7 + H7) = 10

These ar e the first two answer digits. Next, add 3 to onehalf itself.

3 + = 4The number to it s left is 7, which is odd. Therefore add 5.

4 + 5 = 9This is the third answer digit.

1 0 9Th e next digit is 8, and there is an odd digit to it s left.

8 + t(8) + 5 = 8 + 4 + 5 = 1760

Record th e 7 and add the 1 to th e preceding 9, which be comes 10. Record th e 0 and add 1 to the answer digit pre-ceding it .

0 + 1 = 1Th e answer digits ar e now

1 1 0 7The units digit of the given number is even; therefore, theunits digit of th e answer is 0.

1 1, 0 7 0 Answer

..


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28MULTIPLYING BY 25

Rule: Move the decimal point of the given numbertwo places to the right and divide by 4.

What_ being done here is to substitute multiplication ofa two-d1g1t number with division by a single digit, 4.Multiply 649,372 by 25 .First, move the decimal point two places to th e right.649,372.00 becomes 64,937,200.-ext, divide the result by 4.

64, 937,200 + 4 = 1 6, 2 3 4, 3 0 o Answer

62

29MULT IPLYING BY 52

Rule: Move th e decimal point of the given number twoplaces to the right and divide by 2. Add twicethe original number to the result.

Suppose we wanted to find the yearly salary of someoneearning $117 pe r week. Since there ar e 52 weeks in theyear , the problem becomes

117 X 52Move the decimal point of th e given number two places tothe right.

117.00 becomes 11,700-ivide by 2.11 , 700 + 2 = 5,850.To this add twice th e original number.2 X 117 = 234; 5,850+ 2346,084

Thus th e yearly salary of someone earning $117 a week is$6, 084.

63


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30MULTIPLYING A TWO-DIGIT NUMBER BY 95

Here 1s a case where a series of short-cut methodseach capable of being done mentally, are strung t o g e t h ~ rinto one unified short-cut method.

Rule: Subtract 5 from the given number and affix twozeros to the result. This will be called the partial product. Next, subtract the given numberfrom 100 and multiply the result by 5. Addthis product to the partial product to obtainthe final answer.

This can be best demonstrated by trying the example:95 X 73.First subtract 5 from the given number and affix twozeros to the result.

6,800(This is the partial product.) Next, subtract the givennumber from 100 (Shor t Cut 66).

100 - 73 = 27Multiply by 5 (Short Cut 11).

27 X 5 = 135(Note that the tens digit and the units digit of this productare always the tens digit and the units digit of the finalanswer.)

Fmally, add this product to the previously determinedpartial product.

6, 800 + 135 = 6, 935 Answer

64

31

MULTIPLYING BY 125Rule: Move the dec1mal pomt of the given number

three places to th e right and divide by 8.Dividing by 8 may not seem to be much of a short cu t

at first, but a simple application of the method will proveit s worth.Multiply 1,483 by 125.The usual multiplication process would require twelvemultiplication steps plus many steps in addition. Th e shortcu t method uses one step in division. First, move th edecimal point of th e given number three places to the right.

1,483.000 becomes 1,483,000.-ext, divide by 8. Division by 8 can be simplified by di viding th e given number by 2, then dividing the quotient by2, and finally dividing the second quotient by 2. This third

quotient is the final answer. Thus, 1,483,000 can be mentally divided by 2, giving us 741,500. Inspection shows that741,500 can once again be easily divided by 2, giving370,750. Each time we halve the given number, the divisor8 must also be halved.

8/ 2 = 4; 4/ 2 = 2To obtain the product we ar e looking for, we need merelydivide 370,750 by 2.

370,750 + 2 = 185,375 AnswerNaturally, the same answer would have been obtained bydividing by 8 directly.

1 , 483,000 8 = 185,375 Answer

65


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32MULTIPLYING TWO TWO-DIGIT NUMBERS WHENBOTH END IN 5 AND ONE TENS DIGIT IS ODDWHILE THE OTHER IS EVEN

Rule: To the product of th e tens digit add one-halftheir su m (ignoring the fraction t). Affix 75to the result.

This short cut will be tried with the numbers 75 and 45.The product of the tens digits is

7 X 4 = 28One-half the sum of the tens digits (neglecting is

!( 7 + 4) = 5The sum of these two numbers is 33. Affix 75.

3,375Thus

75 X 45 = 3,375A word of caution about "affixinganumber." This merely

means the number is attached or tagged on at the beginningor end of a group of numbers; it does not mean the numberis to be added to another number. -

66

33MULTIPLYING TWO TWO-DIGIT NUMBERS WHENBOTH EKD IN 5 AND THEIR TENS DIGITS AREEITHER BOTH ODD OR BOTH EVEN

Rule: To the product of the tens digits add one-halftheir sum. Affix 25 to the result.

Although to use this shor t-cut method both tens digitsmust be either odd or even, they need not be equal.

I f we ar e asked to multiply 65 by 45, we observe, first,that both tens digits ar e even and this method may be used.The product of the tens digits is 6 x 4 = 24. To this, onehalf the su m of th e tens digits is added.

6 + 4 ::: 10; ! X 10 = 5 24 + 5 = 29Affix 25.2,925Thus

65 X 45 2, 925 Answer

67

. . . . " ' .... ..


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34MULTIPLYING TWO TWO-DIGIT NUMBERS WHOSETENS DIGITS ARE BOTH 5 AND WHOSE UNITSDIGITS ARE BOTH ODD OR BOTH EVEN

Rule: Add one-half th e su m of th e units digits to 25.Affix th e product of the units digits to th e result.If th e product is less than 10, precede it witha zero .

I f we ar e asked to multiply 52 by 58 we se e that th eunits digits are both even and t h e r e f o r ~ this short cut canbe used. The sum of th e units digits is 10 and one-halfthis is 5. '

25 + 5 = 30Multiply th e units digits.

2 X 8 = 16Affix this to the 30 obtained above.

3, 016 AnswerSuppose we ar e asked to multiply 51 by 57. This time th eunits digits ar e both odd. Again th e short cut is applicable.One-half th e su m of th e digits is 4; with this added to 25is. 29. However, in this case th e product ofuruts d1g1ts lS 7, which is less than 10; therefore a zeroprecedes the product before it is affixed to the 29.

2. 907 Answer

68

35MULTIPLYING TWO TWO-DIGIT NUMBERS WHOSETENS DIGITS ARE BOTH 5 AND ONE UNITS DIGITIS ODD WHILE THE OTHER IS EVEN

Rule: Add one-half the sum of the units digits to 25,ignoring th e fraction Add the product of th eunits digits to 50 and affix the result to the sumobtained in the first step.

In this case we need not worry whether the product ofth e units digits is greater or less than 10 since it iseventually added to 50.Let us find the product of 54 and 59. One units digit isodd, while the other one is even. One-half the su m of th eunits digits is 6t. Ignoring th e fraction and adding this to25. we obtain25 + 6 = 31The product of th e units digits is 36. This is added to 50a nd th e su m affixed to 31.

50 + 36 = 863 .1 86 Answer

69



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36MULTIPLYING TWO TWO-DIGIT NUMBERS WHOSETENS DIGITS ARE BOTH 5 AND WHOSE UNITSDIGITS ADD TO 1 0

Rule: Affix the product of the units digits to 30. I fthe product is less than 10, precede i t with azero.

Multiply 53 by 57.The units digits, 3 and 7, total 10 so that this short-cutmethod can be used. Th e first two digits of the answer are30. Th e product of the units digits is3 X 7 = 21

Affix this to 30, resulting in the product:3,021

Through an interesting property of numbers, this sameshort cut can be applied to numbers of more than two di gits .Th e short cut for multiplying numbers in their teens willbe used as an example.Multiply 152 by 158.Imagine just for this example that 52 and 58 may eachbe considered as if they were units digits. In actuality,only the 2 of the first number and only the 8 of the secondnumber ar e the units digits. But what happens i f we treat52 and 58 as units digits? The rule for multiplying teennumbers (Short Cut 25) requires adding the units digit ofone number to the other number . This provides the firsttwo digits of the answer. The product of the units digitsgives the units digit of the answer with any tens digit beingadded to the previously determined sum. Now our "teen"

70

number is 152, and the units digit of th e other number is58. Therefore their sum is

152 + 58 = 210The product of the "units" digits is

52 X 58 = 3,016Remember, we are treating the last two di gits as the unitsdigits in this example. Therefore 1 and 6 ar e the final twodigits in the answer, and the 30 is added to the prevwus sum .

210 + 30 :: 240We now have the result.

152 x 158 = 24.016 AnswerThis is just one way in which a short cut of apparentlylimited application may have it s usefulness enhanced. Byr edefining ou r terms and following through correctly, al mos t any short-cut's area of application may be broadened.

Careful practice and a working knowledge of the intricaciesof numbers as discussed throughout this book ar e al l thatis necessary.

72 SHORT CUTS IN MULTIPLICATION


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Practice Exercises fo r Short Cuts 27 through 361) 713 X 52 =2) 29,621 X 125 =3) 6,104 X 15 =4) 51 X 59 =5) 53 X 566) 8,298 X 25 =7) 65 X 75 =8) 64 X 95 =9) 3,871 X 125 =

10) 52 X 54 =11) 81,927 X 25 =12) 25 X 65 =13) 144 X 52 =14) 92 X 95 =15) 54 X 56 =

NUMBERS BEGINNING OR ENDING IN 9

All numbers ending in 9 ar e one less than a multiple of10. All numbers beginning with 9 ar e some power of 10less than a number beginning with 10. These two character-istics of numbers beginning or ending in 9 ar e used to goodadvantage in the short cuts that follow.Fo r example, i f we increase a number ending in 9 byone, the units digit of the new number is zero. Therefore,we have one less digit to multiply, and a simple subtractionrestores the original multiplier. When a number beginswith 9, it can also be increased easily to a simpler form.Thus 942 can be changed to 1,042 by adding 100 (which is10 x 1 0). Although the new number has four digits, one ofthem is zero and the other is one; both ar e much simplermultipliers than 9.

73


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37MULTIPLYING BY 19

Rule: Double the given number and affix a zero to theresult. Subtract th e given number.

Example: 7,390,241 x 19.Double the given number.

2 X 7,390,241 = 14,780,482Affix a zero and subtract the given number.

1 4 7, 8 0 4, 8 2 07, 3 9 0, 2 4 1

1 4 0, 4 1 4, 5 7 9 Answer

74

38MULTIPLYING BY 99

Rule: Move the decimal point of the given number twoplaces to the right an d subtract the given number.

Multiply 1,152 by 99. .Move the decimal point two places to the nght an d sub-tract th e given number from the result.

1,152.00 becomes 115,200.-15,2001,152114, 048 Answer

75


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Rule: Move the decimal point of th e given numberthree places to the right and subtract thegiven number .

Example: 1,152 x 999.Move the decimal point three places to the right.1,152.000 becomes 1,152,000.-ubtract the given number.

1,152,0001,1521,150,848 Answer

76

40MULTIPLYING BY A NUMBER CONSISTING ONLYOF NINES

Rule: Move th e decimal point of the given number tothe right as many places as there ar e nines inthe multiplier. Then subtract the given number.

Multiply 73 by 9, 999,999.There ar e seven nines in the multiplier; therefore the

decimal point in th e given number wlll be moved sevenplaces to the right.

73.0000000 becomes 730,000,000.Subtract the given number.

730,000,00073

729,999,927 Answer

77


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41MULTIPLYING TWO TWO-DIGIT NUMBERS ENDINGIN 9 AND WHOSE TENS DIGITS ADD TO 10

Rule: Add 9 to the product of the tens digits and af fix 81 to the result.Note that the number 81 is merely attached to the endof the previously determined sum; 81 is not added to the

sum.Fo r example: Multiply 39 by 79.Since the su m of th e tens digits , 3 and 7, is lOthis short-

cut method can be used. The product of the tens digits is3 X 7 = 21

To which 9 is added.21 + 9 = 30

Affix 81 to this sum and obtain the product.3, 081 Answer

42MULTIPLYING BY A TWO-DIGIT MULTIPLE OF 9

Rule: Multiply the given number by one more thanthe tens digit of the multiplier. Move th edecimal point of th e product one place to theright and subtract the original product.

Of course, the usefulness of this short cut is increasedif th e short cuts for multiplying by each of the digits isknown.As an example, multiply 87 by 63 .63 is a multiple of 9 (that is , 9 x 7 = 63}. One moret han the tens digit of the multiplier is 7.Multiply the given number by 7, using Short Cut 13.87 X 7 = 609Move the decimal point one place to the right.609.0 becomes 6,090.

--+Now subtract the original product, 609.

6, 090.- 609.5, 481 . Answer

78 79



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43MULTIPLYING BY ANY TWO-DIGIT NUMBERENDING IN 9

Rule: Move the decimal point of the given number oneplace to the right and multiply by one more thanthe tens digit of th e multiplier. Subtract thegiven number from the result.- - - - - - - - - - - - - - - - - - ~

Multiply 713 by 39 .Move the decimal point of the given number one place tothe right.

713.0 becomes 7,130.-One more than the tens digit of the multiplier is 4. Multiply 7,130 by 4, using Short Cut 10 .

7,130 X 4 = 28,520Subtract the given number.

28,52071327,807 Answer

80

Practice Exercises fo r Short Cuts 37 through 431) 5,803 X 999 =

2) 437 X 39

3) 598,974 X 36 :

4) 1,325 X 19

5) 710 X 99 :

6) 423 X 99,999

7) 29 X 89

8) 53,161 X 19

9) 1,524 X 59 =

10) 69 X 49


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SQUARING NUMBERS

When we speak of "squaring" a number, we mean multiplying th e number by itself. To square 23 we write

23 x 23 (or commonly 232 )Th e process of multiplying a number by itself follows asystematic pattern which lends itself readily to short-cutmethods. The simple rules explained in this section coveran amazingly wide range of numbers. Most of th e shortcuts included here involve two-digit numbers, but a few in volve three- and four-digit numbers. With a little ingenuity, numbers of any size can be squared easily, using theshort cuts that follow as th e basis fo r many others. Butthere is a law of diminishing returns in using larger numbers; then, instead of saving time and labor, the short cu tbecomes merely a "stunt."

Th e squares of numbers play an important role in manyother short-cut methods . By means of th e few very basicmethods in this section, th e range of th e multiplicationproblems which ma y be performed by short-cut methodsbecomes practically unlimited.

82

44SQUARING ANY NUMBER ENDING IN 1! Rule: First, square the number to the left of th e unitsdigit. Then double th e number to th e left of th e

units digit. Affix th e units digit of this result toth e square found in th e first step. I f the result ismore than 9, add the part to the left of the units digitto th e square found in th e first step. Th e unitsdigit of th e answer is always 1.

Consider th e following example: Square 251.Th e number to the left of th e 1 is 25. Using Short Cut45, we find th e square of 25 is 625. Next, twice 25 is 50.Affix th e zero in 50 to 625 and add th e 5 to 625.

625 + 5 = 630To which ar e affixed th e 0 and the units digit (which isalways 1).

63, 001 Answer

83


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45SQUARING ANY TWO-DIGIT NUMBER ENDING IN 5

Squaring a two-digit number ending in 5 is a special case ofthe short cut fo r multiplying any two-digit numbers endingin 5. In this particular case, the tens digits are equal.

Rule: Multiply one more than the tens digit by th eoriginal tens digit and affix 25 to the result.

Fo r example, we shall square 45. First, add 1 to thetens digit.

4 + 1 = 5Next, multiply by the original tens digit.

4 X 5 = 20To this affix 25.

2,025and we have the answer.

45 X 45 = 2,025Remember to merely attach th e 25 to the product: do notadd it to the product.

From this rule we se e that the square of any two-digitnumber ending in 5 always has 5 as it s units d1g1t and 2 asit s tens digit.

84

46SQUARING ANY NUMBER ENDING IN 5

Rule: Multiply the complete number to the left of t1e5 by one more than itself and affix 25 to theresult.

To demonstrate, we shall find th e square of 195. Th ecomplete number to th e left of th e 5 is 19. Ra1sing thisone number higher gives us 20.

20 X 19=380To which 25 is affixed.

38, 025 Answer

85

47

SHORT CUTS IN MULTIPLICATION 87nds digit of the answer Th e thousands digit of thethousa . . .er will be 0 since the hundreds digit of the given num-answ ' h r 1


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SQUARING ANY THREE-DIGIT NUMBER ENDING IN 25The square of 25 is 625. Oddly enough, these are thelast three digits in the square of any three-digit numberending in 25. Since squaring a three-digit number results

in at most si x digits, the problem here is merely to findthe first three digits of the answer.Rule: Th e first two digits (that is, the hundred

thousands digit and the ten-thousands digit) arefound by squaring the hundreds digit of the givennumber and adding to the result one-half thehundreds digit of the given number (ignoring thefraction i f it occurs). I f the result is a onedigit number, then there is no hundred-thousands digit in the answer and the result is theten-thousands digit of the answer. The thousands digit of the answer is 5 i f the hundredsdigit of the given number is odd and 0 if thehundreds digit of the given number is even.Affix 625 to obtain the final answer.

Two illustrative examples will be used to demonstratethe ease with which this short cu t may be used.Examp le No. 1: Square 225

Fi r s t , square the hundreds digit of the given number,to obtain4

To this add one-half the hundreds digit of the given number .4 + 1 = 5

Since the answer is a one-digit number, 5 is the ten-

86

2 is even. To this we afhx 625 to obtam t e Inaer , ,answer.

50,625Example No. 2: Square 725First, square the hundreds digit of the given number.

7 X 7 = 49To th is add one-half of 7 (Ignoring the

49 ... 3 = 52The first digit, 5, is the hundred-thousands digit of theanswer; the second digit, 2, IS the ten-thousands d1g1t ofthe answer. The thousands digit of the answer iS 5, smcethe hundreds digit of the given number is odd. Affix 625to obtain the final answer.

725 x 725 = 5 25,625 Answer



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48SQUARING ANY FOUR-DIGIT !\UMBER E ~ N G 25

Th e explanation for this short-cut method will be madea little clearer if the digits of the given number are assigned letters. The thousands, hundreds. tens, and unitsdigits will be designated A, B, C. and D respectively.

Rule: Square digit A of the given number to obtatn thetentative ten-millions and millions answer dtgit s (if there is only one digit. it is the millionsanswer digit).Double the product of A and B to obtain thehundred- thousands digit of the answer. I f theresult in this and subsequent steps is a twodigit number, the units digit is the answerdigit; the tens digit should be added to the preceding answer digit.

To 5 times A add the square of B. The sumts the ten -thousands digit of the answer.Multiply B by 5. This product is the thousands digit of the answer.Afflx 625 to the answer digits found aboveto obtain the final answer.

This short-cut method will be tried in two illustrative examples.

Example No. 1: Square 2,825. Line the numbers up withtheir respective letters.ABCD2 8 2 5

Square A.2 X 2 = 4

88

The square has only one dtgit; therefore this is ou r tentative millions digtt. Multiply A by B, and double the result.

2 X 8 X 2 : 32Th e 2 is the hundred-thousands digit. Th e 3 1S added tothe 4 obtained in the first step.

3 + 4 = 7Thus 7 is now the millions digit.Add 5 times A to the square of B.

(5 X 2) + (8 X 8) = 74Th e 1 is the ten-thousands digit. Add the 7 to the preceding answer digit.

2 + 7 = 9The 9 becomes the new hundred-thousands dtgit. Stop andrecapitulate what we have:

Millions digitI r Hundred -thousands digitI //--- Ten-thousands digit7 9 4

Multiply B by 5.8 X 5 = 40

Th e 0 is the thousands digit. Add the 4 to the precedmganswer digit.4 + 4 = 8The final ten-thousands dtgit is 8. The prevwus digits, 7and 9, now are also final. Affix 625 to obtain the answer.

90 SHORT CUTS IN MULTIPLICATION SHORT CUTS IN MULTIPLICATION 91


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Th e square of 2, 825 is therefore7, 980, 625 Answer

Example No. 2: Square 7,325. Again the digits will belined up with their respective letters.

ABCD

7 3 2 5Square A.7 X 7 = 49

Since th e answer is a two-digit number the first digit 4' ' 'is th e ten-millions digit and the 9 is the millions digit.Multiply A and B and double th e result.7 X 3 X 2 = 42

Th e 2 is the hundred-thousands digit. Th e 4 is carried toth e left and added to the previous millions digit, 9.

4 + 9 = 13Th e 3 becomes our new millions digit, and th e 1 is addedto the ten-millions digit.

4 + 1 = 5This is th e new ten-millions digit. At this point le t uswrite the digits we have determined:

Ten-millions digit' /Millions digit/ / Hundred-thousands digit

5 3 2Add five times A to the square of B.

(5 X 7) + (3 X 3) = 44One 4 (the units digit) is the ten-thousands digit; the other

4 (the tens digit) is added to th e hundred-thousands digit, 2.4 + 2 = 6

This is the new hundred-thousands digit.Multiply B by 5.

3 X 5 = 15The 5 is th e final thousands digit. Th e 1 is added to th eten-thousands digit.

4 + 1 = 5The 5 is the final ten-thousands digit. Since there is nodigit to carry, th e previous digits become final.Affix 625 to obtain the final answer.

53 , 655,625 Answer

.......... - - ~ ~ .....................


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49SQUARING ANY TWO-DIGIT NUMBER WHOSE TENSDIGIT IS 5

Rule: Add the units digit to 25 and affix the square ofthe units digit to the result. If th e square ofth e units digit is a one-digit number, precedeit with a 0.

Find the square of 53, using this method.Fi r st , add the units digit, 3, to 25.

25 + 3 = 28Next, affix th e square of the units digit to the result.

3 X 3 = 9Since the answer is a one-digit number, place a zero infront of th e 9 before affixing it to th e 28.

2, 809 AnswerAs another example, find th e square of 57.Again, the units digit is added to 25.

25 + 7 = 32Next, square th e units digit

7 X 7 = 49and affix to the previous result.

3, 249 AnsuerThis time th e square of the units digit was a two-digitnumber, and therefore it was not necessary to precede 1twith a zero.

92

50SQUARING ANY NUMBER ENDING IN 9

Rule: Multiply the number to the left of th e 9 by twomore than itself. Affix an 8 to th e result andsubtract twice th e number to the left of the 9.Affix a 1 to the result.

This short cut can be applied to any number, no matterhow many digits it has, so long as th e units digit is 9. Ofcourse, as the number gets larger, multiplying th e twonumber s of th e first step will become cumbersome unlessa shor t cut can be used. However, most two- and threedigit numbers ending in 9 ca n be readily squared, once afacility with the other short-cut methods has been achieved.

Exa mple: Square 149.Th e number to the left of th e 9 is 14. Two more thanth is is 16. Multiply 14 by 16. (Short Cu t 53 can be used

here).14 X 16 = 225 - 1 = 224

Af fix 8.2, 248

Subtract twice the number to th e left of th e 92, 248 - (2 X 14) = 2,248 - 28 = 2,220

Affix a 1 to obtain the final answer.22, 201 Answer

93


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51SQUARING ANY NUMBER CONSISTING ONLY OF NINES

Rule: Write one less 9 than there is in the given num-ber . Follow this with an 8. Then write as manyzeros as the nines previously written. Finally,write a 1 as the units digit.

This method is purely mechanical and requires nothingmore than being able to count the nines in the given number.Square 9, 999.There ar e four nines; therefore write three nines as

the fir

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