The secrets of mental math free download

But you will also learn to view math as an activity that can actually be fun. But as you will learn from Secrets, there are often several ways to solve the same problem.

Large problems can be broken down into smaller, more manageable components. We look for special features to make our problems easier to solve. These strike me as being valuable life lessons that we can use in approaching all kinds of problems, mathematical and otherwise. Many people are convinced that lightning calculators are prodigiously gifted.

Maybe I was born with some curiosity about how things work, whether it be a math problem or a magic trick. But I am convinced, based on many years of teaching experience, that rapid math is a skill that anyone can learn.

And like any worthwhile skill, it takes prac- tice and dedication if you wish to become an expert. But to achieve these results efficiently, it is important that you practice the right way. Let me show you the way!

Mathemagically, Dr. After practicing the meth- ods in this book for just a little while, your ability to work with numbers will increase dramatically.

With even more practice, you will be able to perform many calculations faster than some- one using a calculator.

But in this chapter, my goal is to teach you some easy yet impressive calculations you can learn to do immediately. Did you get ? Now before you get too excited, I have shown you only half of what you need to know. If you got the answer , then give yourself a pat on the back. You are on your way to becoming a mathemagician. You will be amazed at the reaction you get. Whether or not you decide to reveal the secret is up to you! Welcome back. How do I multiply numbers by twelve, or thirteen, or thirty-six?

In Chapters 2, 3, 6, and 8, you will learn meth- ods for multiplying together just about any two numbers. Just a handful of techniques is all that it takes to multiply num- bers in your head, quickly and easily. As you probably know, the square of a number is a number multiplied by itself. Later, I will teach you a simple method that will enable you to easily calculate the square of any two-digit or three-digit or higher number.

To square a two-digit number that ends in 5, you need to remember only two things. The answer begins by multiplying the first digit by the next higher digit. The answer ends in The answer begins the same way that it did before the first digit multiplied by the next higher digit , followed by the prod- uct of the second digits.

How does it end? Remember that to use this method, the first digits have to be the same, and the last digits must sum to Can we use this method to multiply twenty-two and twenty-three? But in Chapter 8, I will show you an easy way to do problems like this using the close-together method.

Not only will you learn how to use these methods, but you will understand why these methods work, too. Instead of subtracting , subtract We will explain how to quickly determine the 13 in Chapter 1. Using a little bit of mathematical magic, described in Chapter 9, you will be able to instantly compute the sum of the ten num- bers below. The answer, , has appeared elsewhere in this chapter. More tricks for doing math on paper will be found in Chapter 6.

We will discuss strategies for calculating sales tax, dis- counts, compound interest, and other practical items in Chapter 5, along with strategies that you can use for quick mental esti- mation when an exact answer is not required. This will be handy in and out of the classroom.

Using an easy-to-learn system for turning numbers into words, you will be able to quickly and easily memorize any numbers: dates, phone numbers, whatever you want. Speaking of dates, how would you like to be able to figure out the day of the week of any date?

I will show you this in more detail later, but here is a simple way to figure out the day of January 1 for any year in the twenty-first century. First familiarize yourself with the following table. Take the last two digits of the year, and consider it to be your bill at a restaurant. You can compute this by cutting the bill in half twice, and ignoring any change. To figure out the day of the week, subtract the biggest mul- tiple of 7 0, 7, 14, 21, 28, 35, 42, 49,.

For more details that will allow you to compute the day of the week of any date in history, see Chapter 9. Are you ready to learn more magical math? Well, what are we waiting for? By adding and subtracting numbers this way, I found that I could call out the answers to math problems in class well before my classmates put down their pencils.

In this chapter you will learn the left-to-right method of doing mental addition and subtraction for most numbers that you encounter on a daily basis.

These mental skills are not only important for doing the tricks in this book but are also indis- pensable in school, at work, or any time you use numbers. Soon you will be able to retire your calculator and use the full capac- ity of your mind as you add and subtract two-digit, three-digit, and even four-digit numbers with lightning speed. When you compute the answer from right to left as you probably do on paper , you generate the answer backward. If you are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right.

But with practice you will find that it is the most natural and efficient way to do mental calculations. With the first set of problems—two-digit addition—the left- to-right method may not seem so advantageous. But be patient. If you stick with me, you will see that the only easy way to solve three-digit and larger addition problems, all subtraction prob- lems, and most definitely all multiplication and division prob- lems is from left to right.

The sooner you get accustomed to computing this way, the better. Two-Digit Addition Our assumption in this chapter is that you know how to add and subtract one-digit numbers.

We will begin with two-digit addition, something I suspect you can already do fairly well in your head. It also illustrates a fundamental principle of mental arithmetic—namely, to sim- plify your problem by breaking it into smaller, more manage- able parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success—simplify, simplify, simplify. The easiest two-digit addition problems are those that do not require you to carry any numbers, when the first digits sum to 9 or below and the last digits sum to 9 or below.

While you need to be able to read and understand such diagrams as you work your way through this book, our method does not require you to write down anything yourself. With practice, however, you will begin to see and hear these numbers in your mind, and car- rying numbers when you add will come automatically. Was that easier? If you would like to try your hand at more two-digit addition problems, check out the set of exercises below.

The answers and computations are at the end of the book. After each step, you arrive at a new and simpler addition problem. The goal is to keep simplifying the problem until you are just adding a one-digit number.

As you simplify the problem, the problem gets easier! Six plus one equals seven, so my next problem is seven hundred and twenty-three plus fifty-nine, and so on. When first doing these problems, practice them out loud. Rein- forcing yourself verbally will help you learn the mental method much more quickly. In my mind the problem sounds like this: plus is plus 34 is plus 4 is However, with this particular problem you have the option of using an alternative method.

It really does not matter which number you choose to break up, but it is good to be consistent. That way, your mind will never have to waste time deciding which way to go.

Since most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calcu- lators, or the mnemonics taught in Chapter 7. In many addition problems that arise in practice, especially within multiplication problems, one or both of the numbers will end in 0, so we shall emphasize those types of problems.

These problems are easy because the nonzero digits overlap in only one place, and hence can be solved in a single step. Where digits overlap in two places, you require two steps. Answers can be found in the back of the book. The German math- ematician Carl Friedrich Gauss — was one such child. He often boasted that he could calculate before he could speak. As a ten-year-old student, Gauss was presented the following math- ematical problem: What is the sum of numbers from 1 to ?

To the astonishment of everyone, including the teacher, young Carl got the answer not only ahead of everyone else, but computed it entirely in his mind.

But if you con- tinue to compute from left to right and to break down problems into simpler components, subtraction can become almost as easy as addition. Two-Digit Subtraction When subtracting two-digit numbers, your goal is to simplify the problem so that you are reduced to subtracting or adding a one-digit number.

Subtract the rounded number, then add back the difference. Just use the rule above to decide which method will work best. Simply sub- tract one digit at a time, simplifying as you go.

Since you subtracted by 8 too much, did you add back 8 to reach , the final answer? Did you notice? But you have sub- tracted too much.

The trick is to figure out exactly how much too much. At first glance, the answer is far from obvious. To find it, you need to know how far is from Quick, how far from are each of these numbers? We say that 43 is the complement of 57, 32 is the com- plement of 68, and so on. Now you find the complements of these two-digit numbers: 37 59 93 44 08 To find the complement of 37, first figure out what you need to add to 3 in order to get 9.

The answer is 6. Then figure out what you need to add to 7 to get The answer is 3. Hence, 63 is the complement of The other complements are 41, 7, 56, Notice that, like everything else you do as a mathemagician, the complements are determined from left to right. As we have seen, the first dig- its add to 9, and the second digits add to An exception occurs in numbers ending in 0—e. What do complements have to do with mental subtraction?

Well, they allow you to convert difficult subtraction problems into straightforward addition problems. But then, having subtracted too much, you needed to figure out how much to add back. Using complements gives you the answer in a flash. How far is from ? The same distance as 68 is from If you find the complement of 68 the way we have shown you, you will arrive at Add 32 to , and you will arrive at , your final answer.

Fortunately, my parents ignored that advice. I was also lucky to have some incredibly patient teachers in my first few years of school. It might have been my short attention span that motivated me to develop quick ways to do arithmetic. In this chapter you will learn how to multiply in your head one-digit numbers by two-digit numbers and three-digit num- bers.

You will also learn a phenomenally fast way to square two-digit numbers. I sometimes wonder whether we were not cheated in school; these methods are so simple once you learn them. There is one small prerequisite for mastering the skills in this chapter—you need to know the multiplication tables through ten. In fact, to really make headway, you need to know your multiplication tables backward and forward.

For those of you who need to shake the cobwebs loose, consult the multiplication chart below. You will do vir- tually all the calculations in this chapter from left to right as well. This is undoubtedly the opposite of what you learned in school. For one thing, you can start to say your answer aloud before you have finished the calculation. That way you seem to be calculating even faster than you are! Then add plus 14 left to right, of course to arrive at , the correct answer.

We illustrate this procedure below. At first you will need to look down at the problem while doing the calculation. With practice you will be able to forgo this step and compute the whole thing in your mind.

The answer is Note: If you are wondering why this process works, see the Why These Tricks Work section at the end of the chapter. Try doing them in your head before looking at how we did it. Another especially easy type of mental multipli- cation problem involves numbers that begin with five. When the five is multiplied by an even digit, the first product will be a mul- tiple of , which makes the resulting addition problem a snap.

The addition problem is slightly harder because it involves carrying a number. Rounding Up You saw in the last chapter how useful rounding up can be when it comes to subtraction. The same goes for multiplication, especially when you are multiplying numbers that end in eight or nine. As it is, you may prefer to stick with the addition method. Per- sonally, for problems of this size, I use only the addition method because in the time spent deciding which method to use, I could have already done the calculation!

So that you can perfect your technique, I strongly recommend practicing more 2-by-1 multiplication problems. Below are twenty problems for you to tackle. I have supplied you with the answers in the back, including a breakdown of each component of the multiplication.

Cal- culate mentally, then check your answer with a calculator. Once you feel confident that you can perform these problems rapidly in your head, you are ready to move to the next level of mental calculation. If this problem gave you trouble, you might want to review the addition material in Chapter 1.

Since you do not need to carry any numbers, it is easy to add 42 to to arrive at the total of There is no magic secret to remembering that first number, but with practice I guarantee you will improve your concentration, and holding on to numbers while performing other functions will get easier.

This is okay at first. But try to break the habit so that eventually you are holding the problem entirely in memory.

This is because you do not have to carry any numbers and the thou- sands digit does not change. As an added bonus, by quickly saying the first digit, it gives the illusion that you computed the entire answer immediately! The difficult part comes in holding the pre- liminary answer in your head while computing the final answer.

Some- times at this stage I will start to say my answer aloud before fin- ishing. The next two problems require you to carry two numbers each, so they may take you longer than those you have already done. I can assure you from experience that doing mental calculations is just like riding a bicycle or typing. I can still recall where I was when I discovered how to do it. I was thirteen, sitting on a bus on the way to visit my father at work in downtown Cleveland.

It was a trip I made often, so my mind began to wander. I noticed that the products were getting smaller, and their differ- ence from was 1, 4, 9, 16, 25, 36,. Next I tried numbers that add to 26 and got similar results. Just as before, the distances these products were from was 12, 22, 32, 42, and so on see table below. There is actually a simple algebraic explanation for this phe- nomenon see Why These Tricks Work, page Then I realized that this pattern could help me square num- bers more easily.

Suppose I wanted to square the number Almost done! Can squaring a two-digit number be this easy? Yes, with this method and a little practice, it can. And it works whether you initially round down or round up. In fact, for all two-digit squares, I always round up or down to the nearest multiple of So if the number to be squared ends in 6, 7, 8, or 9, round up, and if the number to be squared ends in 1, 2, 3, or 4, round down. With this strategy you will add only the numbers 1, 4, 9, 16, or 25 to your first calculation.

Since you will always round up and down by 5, the numbers to be multiplied will both be multiples of Hence, the multiplication and the addition are especially simple. Even large numbers are not to be feared. But, in fact, these are even easier because they allow you to round up to Try it yourself, and then check how we did it. No matter where he went, Colburn met all challengers with speed and precision. Answered in twenty seconds: , days, 15,, hours.

How many sec- onds in eleven years? Answered in four seconds; ,, He died at a youthful thirty-five. Some people may find the theory as interesting as the application. Fortunately, you need not understand why our methods work in order to under- stand how to apply them.

All magic tricks have a rational expla- nation behind them, and mathemagical tricks are no different. It is here that the mathemagician reveals his deepest secrets! In this chapter on multiplication problems, the distributive law is what allows us to break down problems into their com- ponent parts. To understand it intuitively, imagine having 7 bags, each containing 42 coins, 40 of which are gold and 2 of which are sil- ver.

How many coins do you have altogether? There are two ways to arrive at the answer. Notice that the numbers 7, 40, and 2 could be replaced by any numbers a, b, or c and the same logic would apply. As for squaring, the following algebra justifies my method.

I experienced my first public performance in eighth grade, at the fairly advanced age of thirteen. Many mathemagi- cians begin even earlier. Zerah Colburn — , for exam- ple, reportedly could do lightning calculations before he could read or write, and he was entertaining audiences by the age of six! When I was thirteen, my algebra teacher did a problem on the board for which the answer was How did you do it? I was thrilled. I actu- ally believed I had discovered something new.

Still, the fact that I had discovered it for myself was very exciting to me. You, too, can impress your friends or teachers with some fairly amazing mental multiplication.

At the end of the last chapter you learned how to multiply a two-digit number by itself. In this chapter you will learn how to multiply two differ- ent two-digit numbers, a challenging yet more creative task.

You will then try your hand—or, more accurately, your brain—at three-digit squares. You do not have to know how to do 2-by-2 multiplication problems to tackle three-digit squares, so you can learn either skill first. When multiplying two-digit numbers, however, you can use lots of different methods to arrive at the same answer. For me, this is where the fun begins. The Addition Method To use the addition method to multiply any two two-digit num- bers, all you need to do is perform two 2-by-1 multiplication problems and add the results together.

So how do you decide which number to break up? I try to choose the number that will produce the easier addition problem. In most cases—but not all—you will want to break up the number with the smaller last digit because it usually produces a smaller sec- ond number for you to add. I did. Once all future dates are taken care of, we can look back into the past and determine the days of the week for any date in the s or any other century. Every year is assigned a code number, and for that year code happens to be 0 see page Now, to calculate the day of the week, you simply add the month code plus the date code plus the year code.

How about November 18, ? Now since the week repeats every seven days, we can sub- tract any multiple of 7 from our answer 7, 14, 21, 28, 35,. Hence November 18, , occurs on Saturday. How about ? Well, what happens to your birthday as you go from one year to the next? If there are days between your birthdays, then it will shift forward by two days. Hence, for we cal- culate the day of the week just as before, but now we use a year code of 1. Next, is a leap year. Leap years occur every four years, so , , , ,.

This makes the sub- sequent addition problem much easier. Suppose you have a two-digit number whose digits add up to 9 or less. To multiply this number by 11, merely add the two digits together and insert the total between the original two digits. If you place the 6 between the 4 and the 2, you get , the answer to the problem! The 2-by-2 multiplication problems really do not get any tougher than this. But before you read on, practice the addition method on the follow- ing multiplication problems.

But you multiplied by too much. How much? So subtract 46 from to arrive at , the final answer. Go through them and say the steps to yourself or even out loud to reinforce your thoughts. Not only do I use the subtraction method with numbers that end in 8 or 9, but also for numbers in the high 90s because is such a convenient number to multiply. We know the answer will be in the s. The difference between 40 and 78 is Now take the complement of 38 to get Hence the answer is Try this one.

As you may have realized, you can use this method with any subtraction problem that requires you to borrow a number, not just those that are part of a multiplication problem. All of this is further proof, if you need it, that complements are a very pow- erful tool in mathemagics.

Master this technique and, pretty soon, people will be complimenting you! You use it when one of the numbers in a two-digit multiplica- tion problem can be factored into one-digit numbers. To factor a number means to break it down into one-digit numbers that, when multiplied together, give the original num- ber. In most cases, I like to use the larger factor in solv- ing the initial 2-by-1 problem and to reserve the smaller factor for the 3-by-1 component of the problem. Factoring results in a 2-by-2 multiplication problem being simplified to an easier 3-by-1 or sometimes 2-by-1 multiplica- tion problem.

The advantage of the factoring method in mental calculation is you do not have to hold much in memory. By the way, the reason we can shift parentheses in the second step is the associative law of multiplication. With the factoring method, you have just two multiplication problems: a 2-by-1 and a 3-by-1, and then you are done. The factoring method is usually easier on your memory.

Our next example illustrates that it sometimes pays to factor the numbers in an order that exploits this situation. I call such numbers friendly products. The following list should help. With practice you will be able to nose out friendly products more often, and the list will become more meaningful.

You have mastered 2-by-2 multiplication and now have all the basic skills you need to be a fast mental calculator. All you need to become a lightning calculator is more practice!

Many of the following exercises can be solved by more than one method. Try computing them in as many ways as you can think of, then check your answers and computations at the back of the book. Our answers suggest various ways the problem can be mathemagically solved, starting with what I think is the easiest method. You can do these problems now for practice, and refer back to them when they are used in the larger problems.

Just as you square two-digit numbers by round- ing up or down to the nearest multiple of 10, to square three- digit numbers, you round up or down to the nearest multiple of These last problems are not terribly hard because there is no real addition involved.

Moreover, you know the answers to 62 and 72 by heart. Tack on two 0s to arrive at 98, Then add the square of If you need time to compute , repeat the number 98, to yourself a few times before you go on. The farther away you get from a multiple of , the more difficult squaring a three-digit number becomes. But this will not work for every problem. The key here is to repeat , to yourself several times. Then square 36 to get 1, in the usual way.

The hard part comes in adding 1, to , Do it one digit at a time, left to right, to arrive at your answer of , Take my word that as you become more familiar with two-digit squares, these three-digit problems get easier. The first problem is deciding what numbers to multiply together. Clearly one of the numbers will be , and the other number will be in the s. But what number? You can com- pute it two ways: 1. The hard way: the difference between and is 37 the complement of Subtract 37 from to arrive at The easy way: double the number 63 to get , and take the last two digits to give you Because both numbers are the same distance from , their sum must be twice , or One of your numbers is , so the other must be In a later chapter, you will learn a memory system that will make remembering such numbers much easier.

Adding , or 1,, gives you , If you got it right the first time, take a bow! The paradox has come to be known as the Monty Hall problem, and it goes like this. Monty Hall allows you to pick one of three doors; behind one of these doors is the big prize, behind the other two are goats.

You pick Door Number 2. Now, in his tantalizing way, Monty gives you another choice. What should you do? Assuming that Monty is only going to reveal where the big prize is not, he will always open one of the consolation doors. This leaves two doors, one with the big prize and the other with another consolation. The odds are now for your choice, right?

The odds that you chose correctly the first time remain 1 in 3. The probability that the big prize is behind the other door increases to 2 in 3 because the probability must add to 1. Thus, by switching doors, you double the odds of winning! The problem assumes that Monty will always give a player the option to switch, that he will always reveal a nonwinning door, and that when your first pick is correct he will choose a nonwinning door at ran- dom. Think of playing the game with ten doors and after your pick he reveals eight other nonwinning doors.

Here, your instincts would probably tell you to switch. People confuse this problem for a variant: if Monty Hall does not know where the grand prize is, and reveals Door Number 3, which happens to contain a goat though it might have contained the prize , then Door Number 1 has a 50 percent chance of being correct.

They were all wrong. Recall that the cube of a number is that number mul- tiplied by itself twice. As you will see, this is not much harder than multiplying two-digit numbers. Just like with squaring two-digit num- bers, I choose d to be the distance to the nearest multiple of ten. Try the cube of How many times a week are you confronted with situations that call on you to evenly divide things, such as a check at a restaurant?

The ability to divide in your head can save you the inconve- nience of having to pull out a calculator every time you need to compute something. With mental division, the left-to-right method of calculation comes into its own. This is the same method we all learned in school, so you will be doing what comes naturally. I remember as a kid thinking that this left-to-right method of division is the way all arithmetic should be done. I have often speculated that if the schools could have figured out a way to teach division right-to-left, they probably would have done so!

Knowing that, we first determine the largest multiple of 10 that can be multiplied by 7 whose answer is below At this point we can actually say the number 20 since that part of our answer will certainly not change. Therefore, your answer is 80 something. To find out, subtract from for a remainder of The more you calculate, the simpler the problem becomes.

Thus we must first find the largest multiple of that can be squeezed into We can simplify our problem by doubling both numbers. But for greater dramatic effect, you may prefer, as I do, to hold the answer on your fingers and say it all together at the end. In that case, you may run into problems remembering digits greater than five if, like most of us, you have only five fin- gers on each hand.

You already know that to represent numbers 0 through 5, all you have to do is raise the equivalent number of fingers on your hand. With three-digit answers, hold the hundreds digit on your left hand and the tens digit on your right.

In that case, since you have only two hands, you will have to say the thousands digit of the answer out loud and use the rule of thumb to remember the rest of the answer.

Naturally, division problems become harder as the number you divide by gets larger. Fortunately, I have some magic up my sleeve to make your life easier. To determine the answer, your first step is to ask how many times 14 goes into Next, subtract from , which is 37 and reduces your problem to dividing 14 into Subtracting 28 from 37 leaves you a remainder of 9.

To figure out the tens digit of the two-digit answer, you need to ask how many times 23 goes into Now you know that the answer is 20 something, and you can say so. Aside from knowing that the answer is 7, you can also compute the remainder right away. Since is 5 more than , the remainder will be 5 less than 62, the divisor. To arrive at the first digit of the answer, you need to figure how many times 54 goes into As promised, I want to share with you a couple of tricks for simpli- fying certain mental division problems.

These tricks are based on the principle of dividing both parts of the problem by a com- mon factor. Therefore, when using this method, you must always express the answer in fractional form. Check the end of the book for answers and explanations. Only the starting point varies. You can figure out the starting point in a flash by multiplying. The rest follow in a similar way. However, keep your eyes peeled for ways of simplifying such problems. If the denominator of the fraction is an even number, you can simplify the fraction by reducing it in half, even if the numerator is odd.

For example: 9 4. Although the sevenths sequence previously 4. When the divisor ends in 5, it almost always pays to double the problem, then divide by For example, 29 58 5. Wher- ever appropriate, simplify the fraction before converting it to a decimal.

We end this chapter with a brief discussion of how to determine whether one number is a factor of another number. Being able to find the factors of a number helps us simplify division prob- lems and can speed up many multiplication problems. Being able to factor these numbers quickly is very handy. And besides, I think some of the rules are just beautiful. All you need to do is to check if the last digit is even. If the last digit is 2, 4, 6, 8, or 0, the entire number is divisible by 2.

To test whether a number is divisible by 4, check if the two- digit number at the end is divisible by 4. The number 69, is not a multiple of 4 because 46 is not a multiple of 4. The reason this works is because 4 divides evenly into and thus into any multiple of Thus, since 4 divides evenly into 57,, and 4 divides into 52, we know that 4 divides evenly into their sum, 57, Likewise, since 8 divides into , to test for divisibility by 8, check the last three digits of the number.

For the number 14,, divide 8 into Since 27 is a multiple of 3, we know 57, is a multiple of 3. The same amazing rule holds true for divisibility by 9. A number is divisible by 9 if and only if its digits sum to a multiple of 9. Hence, 57, is a multiple of 9, whereas 31,, which sums to 15, is not. The reason this works is based on the fact that the numbers 1, 10, , 1,, 10,, and so on, are all 1 greater than a multiple of 9.

To establish whether a number is divisible by 5 is even easier. Any number, no matter how large, is a multiple of 5 if and only if it ends in 5 or 0. Establishing divisibility by 11 is almost as easy as determin- ing divisibility by 3 or 9. A number is divisible by 11 if and only if you arrive at either 0 or a multiple of 11 when you alternately subtract and add the digits of a number.

The reason this works is based, like the rule for 3s and 9s, on the fact that the numbers 1, , 10,, and 1,, are 1 more than a multiple of 11, whereas the numbers 10, 1,, ,, and so on are 1 less than a multiple of Testing for divisibility by 7 is a bit trickier. If you add or sub- tract a number that is a multiple of 7 to the number you are test- ing, and the resulting number is a multiple of 7, then the test is positive.

I always choose to add or subtract a multiple of 7 so that the resulting sum or difference ends in 0. For example, to test the number , I subtract 42 a multiple of 7 to obtain Next, I get rid of the 0 at the end since dividing by ten does not affect divisibility by seven , leaving me with Then I repeat the process by adding 35 a multiple of 7 , which gives me Therefore, the original number is divis- ible by 7.

Dropping the 0 results in Dropping the two 0s leaves you with 9, which is clearly not a multiple of Therefore, is not a multiple of The rest should be easy for you. Divisibility by 2 1. In this section, we review the basic methods for adding, subtracting, multiplying, dividing, and simplifying fractions.

Those already familiar with fractions can skip this section with no loss of continuity. Multiplying Fractions To multiply two fractions, simply multiply the top numbers called the numerators , then multiply the bottom numbers called the denominators. Try these exercises before going further. First, turn the second fraction upside down this is called the reciprocal then multiply.

Divide these fractions. Now we know that when we multiply any number by 1, the number stays the same. In other words, if we multiply the numerator and denominator by the same number, we get a frac- tion that is equal to the first fraction. If the denominators are equal, then we add the numerators and keep the same denominators. When the denomi- nators are not equal, then we replace our fractions with frac- tions where the denominators are equal. We have illustrated with examples and provided exercises for you to do.

All you really need at this information-gathering stage is a ballpark estimate of what your monthly payments will be. The guesstimation methods described in this chapter will make both these tasks—and many more just like them—much easier.

Addition, subtraction, divi- sion, and multiplication all lend themselves to guesstimation. Like most lightning calcula- tors, Bidder began to try his hand and mind at mental arithmetic as a young lad. Learning to count, add, subtract, multiply, and divide by playing with marbles, Bidder went on tour with his father at age nine. Almost no question was too difficult for him to handle.

At age ten, Bidder mentally computed the square root of ,,, as , in a mere thirty seconds. Riding on his fame, George Bidder entered the University of Edin- burgh and went on to become one of the more respected engineers in England. As late as , in fact, just before his death, Bidder calculated the number of vibrations of light striking the eye in one second, based on the fact that there are 36, waves of red light per inch, and light travels at approximately , miles per second.

Since the exact answer is 14,, our relative error is small. This is what I call a good guesstimation! If you can compute these smaller problems exactly, you can guesstimate the answer to any addi- tion problem. Have you ever gone to the store and wondered what the total is going to be before the cashier rings it up?

Dividing 6 into 58 gives you 9 with a remainder. But the most important com- ponent in this problem is where to place the 9. This tells you the answer is 9, and something. At this point you could bring down the 0 and divide 6 into 40, and so forth. In fact, the actual answer is 9,—darn close! Division on this level is simple. But what about large division problems? Does this player earn thousands every day?

Does he earn tens of thousands every day? Not a bad estimate and not a bad salary! Well, light travels at , miles per second, and the sun is on aver- age 92,, miles away. Then append the two 0s you removed from 93, and you get seconds. The exact answer is You can do better if you round both numbers by the same amount, but in opposite directions. Your guesstimation is off by only 1. When you guesstimate the answer to multiplication problems by rounding the larger number up and the smaller number down, your guesstimate will be a little low.

If you round the larger number down and the smaller number up so that the numbers are closer together, your guesstimate will be a little high. The larger the amount by which you round up or down, the greater your guesstimate will be off from the exact answer. You can see that this multi- plication guesstimation method works quite well. Also notice that this problem is just and that our approximation is just the first step of the squaring techniques. How high can you go with this system of guesstimating multi- plication problems?

As high as you want. You just need to know the names of large numbers. A thousand thousand is a million, and a thousand million is a billion. Hence the answer is roughly billion, since a thousand million is a billion. The square root is used in many science and engineering problems and is almost always solved with a calculator. The following method provides an accurate estimate of the answer. In square root estimation your goal is to come up with a num- ber that when multiplied by itself approximates the original num- ber.

Since the square root of most numbers is not a whole number, your estimate is likely to contain a fraction or decimal point. Your first step is to think of the number that when multiplied by itself comes closest to Since 25 is too high, the answer must be 4 point something. Your next step is to divide 4 into 19, giving you 4. Hence, the square root of 19 lies between 4 and 4. In fact, the square root of 19 rounded to three deci- mal places is 4.

We know 4 squared is 16, which is shy of 19 by 3. We note that this method will always produce an answer that is a little higher than the exact answer. Now you try a slightly harder one. Carrying out the division of 9 into 87 to two decimal places, you get 9. To improve your guesstimate, take the average of 9 and 9. But what about three-digit numbers? Actually, they are not much harder. I can tell you right off the bat that all three-digit and four-digit numbers have two-digit square roots before the decimal point.

And the procedure for comput- ing square roots is the same, no matter how large the number. For instance, to compute the square root of , first find your ballpark figure.

Because 20 squared is and 30 squared is , the square root of must lie between 20 and When you divide 20 into , you get approximately To guesstimate the square root of four-digit numbers, look at the first two digits of the number to determine the first digit of the square root. For example, to find the square root of , consider the square root of So the answer is 80 something.

Now proceed the usual way. Dividing 80 into gives 92 plus a fraction, so a good guesstimate is In practice, you only need to look at the square root of the first two digits of six-digit numbers or the first digit of five numbers. Once you figure out that the square root of 59 lies between 7 and 8, you know your answer is in the s. But you could have been closer, as the fol- lowing trick demonstrates.

Your first step is to round to the nearest large number—in this case, just find the square root of Divide: Average: 5. The exact answer is slightly greater than Not bad! This wraps up the chapter on guesstimation math. A precociously brilliant student, Galois laid the foundation for a branch of mathematics known as group theory. Legend has it that he penned his theory the night before the duel, anticipating his demise and wanting to leave his legacy to the mathematics commu- nity.

The first concerns the theory of equations, the others integral func- tions. After that, I hope some men will find it profitable to sort out this mess. What Galois penned the night before his death were correc- tions and editorial changes to papers that had been accepted by the Academy of Sciences long before. It was after this that Galois became embroiled in political controversy, was arrested, spent time in a prison dungeon, and, ulti- mately, got himself mixed up in a dispute over a woman and killed.

Either multiply the amount by 25, then divide by , or divide the amount by 4 perhaps by cutting the amount in half twice. Perhaps the easiest way to add half of a percent to any dollar amount is to simply cut the dollar amount in half, then turn it into cents.

If you would rather divide by 10 instead of 12, go ahead and do it. You will be calculating 6 6. To add a quarter percent, you can divide the original dollar amount by 4 or cut it in half, twice and turn the dollars into cents.

How would you calculate a sales tax of 7. For example, to calculate 7. We can use this approximation procedure for any sales tax. Thus D is equal to A times the reciprocal of B. Adding these numbers together gives you the total sales tax or an approximate one, if you rounded D to an easier nearby number.

For instance, with 4 28 1 7. To divide a number by 16, divide the number by 4 twice, or divide the number by 8, then by 2. Try to come up with meth- ods for finding the sales tax in the state that you live in. You will find that the problem is not as taxing as it seems! Now suppose that you borrow money, and you have to pay the money back.

About how much will you need to pay each month? Although, actu- ally, what you owe in interest will go down gradually over time. But fortunately you do not need to pay that much extra. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessly. This is the math they never taught you in school. User: Noor Khan. These simple math secrets and tricks will forever change how you look at the world of numbers.

Read Book Download. Mathematics Words Ages 0 and up 31 Publication Date: Posts and Comments Write a new post. Nice thanks for this mental math book, it really did change my POV of numbers! Permalink Comment Comments Are Closed.