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A fun and challenging way for kids to practice basic math skills. Multiplying by 15 60 Multiplying by 25 62 Multiplying by 52 63 Multiplying a Two- Digit Number by 95 64 3l. Multiplying by 65 Multiplying by 19 74 Multiplying by 99 75 Multiplying by 76 Multiplying by a Number Consisting Only of Nines 77 viii 7.

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Multiplying by a Two-Digit Multiple of 9 79 Squaring Any Number Ending in 1 83 Squaring Any Number Ending in 5 85 Squaring Any Number Ending in 9 93 Subtracting a Number from the Next Highest Power of 10 Divisibility by 2 Divisibility by 3 Divisibility by 4 Divisibility by 5 Divisibility by 6 Divisibility by 7 Divisibility by 8 Divisibility by 9 Divisibility by 11 Dividing by 5 Dividing by 15 Dividing by 25 Dividing by 9 Dividing by Factors x 9.

Adding Two Fractions V,.

Do-It-Yourself Short Cuts xi: That anony-mous caveman who chipped the corners off a flat rock andinvented the wheel started this tradition. Most of mans efforts in the past were directed at con-serving 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 atten-tion turned to reducing such laborious tasks as calculating.

WHAT SHORT CUTS ARE Short cuts in mathematics are ingenious little tricks incalculating that can save enormous amounts of time andlabor - not to mention paper - in solving otherwise com-plicated problems, There are no magical 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 are by no means of recent origin; they were knowneven to the ancient Greeks. The supply of short cuts is un-limited. Many are known, and many are yet to be dis-covered. The 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. Integers are numbers consisting of one or more digits. Forexample, 72, is an integer consisting of five digits, 7,2, 9, 5, and 8.

In practice, the word number is applied tomany different combinations of digits ranging from wholenumbers, to fractions, mixed numbers, and decimals. Theword integer, however, applies only to whole numbers. Each digit in a number has a name based on its positionin the number. The number system we are accustomed todealing with is based on the number Each number po-sition in this system is named for a power of The po-sition immediately to the left of the decimal point of anumber is called the units position.


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In the number 1. In fact, any digit that occupies that pOSition is called theunits digit. The next position to the left of the units posi-tion is called the tens position, and any digit occupying thatspace is called the tens digit. In the number Continuing to the left, in order, are thehundreds, thousands, ten-thousands, hundred-thousands,millions pOSitions, and so on.

The positions of the digits to the right of the decimalpoint also have names similar to those to the left. ThepOSition immediately to the right of the decimal point iscalled the tenths pOSition. Notice that the name is tenthsand not tens. In fact, all positions to the right of the deci-mal point end in ths.


The next pOSition to the right of thetenths position is the hundredths position, then the thou-sandths pOSition, and, in order, the ten-thousandths, thehundred-thousandths, the millionths. The positionto the left of the decimal point is always the units position;the one to the right is always the tenths position, no matterwhat digit occupies the space. In addition to the names of the positions as given above,the letters A, B, C, For that reason it is important not only tolearn the various pOSition names but also to gain familiari-ty with the letter notation just mentioned.

Both will be usedfrequently throughout this book. The point is used to separate those numbers that are equalto or greater than 1 from those numbers that are less than1. Even if we write a number without a decimal point, it These three forms are ex-actly equal. In the first case the decimal is omitted butnevertheless is understood to be to the right of the 7. It isalso understood that the only digits that can be placed tothe right of the decimal point without changing the 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 see whythis is an important property of decimals. 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 Also,never confuse the hundreds digit with the hundredths digit. Once you have familiarized yourself with the language,the next step is to develop a routine for learning and mem-orizing the short cuts.

Maximum efficiency can 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 its group.

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It would be pointless to have to refer to thisbook each time you wanted to apply a short cut. We always started from the units digit and worked to theleft. After we got our answer, we reversed the number inour mind and read it from left to right. Not only was theprocess awkward, but the mental gymnastics wasted time. Take this simple example: x 7The product was obtained in the following order: 8, 4, 5, 2Then to read the answer, it became 2,Why cannot answers be obtained in their natural readingorder?

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There is no reason at all why we cannot solve prob-lems just as easily from left to right 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, its advantages will become evident and the time-saving ease with which it can be used will prove its worth.

In this book, the term "first digit" refers to the left-. First, read and reread the Rules as many times as nec-essary at least twice until a general idea of the short-cutmethod is established in your thought. Keep in mind that Try 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. If you dont get all the stepscorrect, go back over them again. You may have to re-read the Rule. Finally, when you are completely satisfied that you havemastered the short cut, try the Practice Exercises.

Theanswers should be written directly in the space provided.

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Try doing intermediate steps mentally. Very soon youllfind that you can solve most problems without paper andpencil. Remember, systematic study and concentration on whatyou are doing are vital to the mastery of each of the short cuts in mathematics. Doyou remember the admonition, never to add dissimilar ob-jects? One must not add 2 oranges to 2 apples unless onewere making fruit salad.