Chapter 13 - Breakdown

It is hardly likely that you would ever multiply a number by ten by putting down the number with ten under it, and multiplying out digit by digit like this:

4 6 3 0

Instead, you know that in order to multiply any number by ten you simply add a zero. If the number has a decimal point in it, you move the decimal point to the right instead of adding the zero.

984 x 10 is 9840.

653.92 x 10 is 6539.2.

Elementary as this is, the principle is basic to many of the short cuts in number work. In many cases, we can save time by multiplying or dividing a number by ten, a hundred, or even a thousand before even beginning work.

In division, of course, you remove a zero (or move the decimal point one place to the left) in order to divide by ten.

2390 divided by 10 is 239.

718.64 divided by 10 is 71.864.

In order to avoid any possible confusion, make sure you understand that any whole number is presumed to have a decimal after it. We shall get more deeply into the subject in the chapter on decimals, but for the moment let's point out that 75 can be considered to be 75.00. Then, if we divide by 10, we move that presumed decimal one place to the left. 75 divided by 10 is 7.5.

Each digit, you remember, increases tenfold in value as it moves one place to the left. So to multiply by a hundred, we add two zeros (ten times ten), or move the decimal point two places to the right.

984 x 100 is 98400.

653.92 x 100 is 65392.

When dividing by a hundred, we also move the decimal point two places—to the left.

984 divided by 100 is 9.84.

653.92 divided by 100 is 6.5392. We would most likely round it off to 6.54.

Undoubtedly none of this is new to you. It is merely a refresher. But the refresher is important, because the more easily and automatically you can think this multiplication or division by ten or a hundred, the more quickly and confidently you will handle the short cuts that involve such division or multiplication as a basic part.

Our second step into the breakdown short cut is through another obvious technique that may well be second nature to you already.

In dealing with many numbers, you probably know already how to multiply by 9 in the "round off and adjust” method. Rather than multiplying by 9, you multiply by 10— and subtract 1.

Compare the two methods:

The working in these two examples is not dramatically different, but they are cited to illustrate a point and to lead into more sophisticated examples. Once again, for the sake of your number sense, try to "feel" the identity of the two expressions above of precisely the same situation.

Just as you probably already knew this special dodge in handling 9, it is likely that you have used in the past the same sort of approach in handling numbers very near 100.

If you have to multiply 238 by 99, surely you would not bother to set up the whole problem and multiply it out line by line. You just subtract one 238 from a hundred 23 8's, as this comparison demonstrates:

Usual way                                Breakdown way

If you were required to multiply by 101, on the other hand, you would simply add one 238 to a hundred 238's. This, after a very moderate amount of practice, you easily do in your head. After a few tries, you should be able to "see" the answer as 24038.

Not very often is your work as extra-simple as multiplying by 99 or 101. But the principle works in a surprising variety of cases, and is the "round off and adjust" special subdivision of our first general short cut: breakdown.

The over-all rule for breakdown is this: break one of your numbers down into two easier-to-handle numbers.

Thus we broke 9 down into 10 and 1.

We broke 99 down into 100 and 1.

We can also—here is where the method becomes far more generally useful—break 45 into 50 less 5. 5, you note, is exactly 1/10 of 50. Or we break 44 down into 40 plus 4— the 4 being exactly 1/10 of 40.

Stop for a moment and try the first example:

Usual way        Breakdown way

It is especially helpful when you can break down a number into two parts of which one is an even fraction of the other, such as 50 and 5. You cannot always do this, of course, which is why we also use other short cuts.

The exact breakdown may well depend on the relationship between the numbers to be multiplied. In some cases one breakdown will make sense, in other cases quite a different breakdown.
Note how it varies in these two cases:

Which breakdown of 18 might you use in the first example? The number 18 can be broken into 12 plus 6 (½ of 12), into 9 plus 9 (two equal parts), into 20 minus 2 (1/10 of 20).

For the first example, the most convenient breakdown of 18 might well be 12 plus 6—because most of us have dealt enough in grosses to know almost by instinct that 12 x 12 is 144. So 18 x 12 is 144 plus ½ of 144 (72)—which we can see as 216.

For the second example, however, most of us could not quickly "see" the answer to 62 x 12. If we break down 18 into 9 plus 9, we can quickly multiply 62 x 9 and then add the answer to itself. Furthermore, we can multiply by 9 using 10 minus 1. This is two-step breakdown. Complex as it may seem at first glance, a very quick and simple way of solving this example would be to handle it as "620 minus 62— doubled."

If we chose the third breakdown, our number work would be surprisingly similar to that involved in the second. 20 minus 2 is identical to 10 minus 1, doubled; only the order of operation is changed.
 
Let us choose the (10 minus 1) doubled breakdown for 18 in this case. Here is how the work should look:

The advantages of this sort of breakdown show up more dramatically, of course, in longer numbers. Try one of the last two breakdowns of 18 on the following problem. Use your pad and pencil:

As with many of the demonstrations, the short-cut nature of the method is not as striking at first sight as you will find it in actual practice. Often you will use almost as many figures, and as many operations. But you are using basically simpler combinations: multiplying by 10 instead of by 9; subtracting instead of doing another digit-by-digit multiplication; doubling instead of adding two lines.

Just for comparison, here is how the 20 minus 2 breakdown for 18 works in the same problem:

In this case, it is presumed that you can jot down twice any figure at sight, and add a 0 at the end to get the effect of multiplying by 20.

There is virtually no limit to the breakdowns you can find. You can break down a number into two parts that add up to the original number (such as 12 plus 6 in 18, or 10 plus 5 in 15) or two parts of which you subtract one from the other to get the number (such as 100 minus 1 for 99, 20 minus 2 for 18, 60 minus 6 for 54).
 
How would you break down 81? Depending on the number you needed to multiply, you could make it 80 plus the original number, or 90 minus 1/10 of the product (since 90 minus 9 is 81, and 9 is 1/10 of 90).

Your proportions need not always be 1/10. They can be ½, 1/3, ¼ or any other convenient fraction. The key is to find a convenient fraction, or there is no sense in using the breakdown method.

See if you can recognize convenient breakdowns for these numbers:

39        26        77        63        125      720

Of 39, we can make 40 minus 1. Of 26, we would make 25 plus 1. In another short cut, incidentally, you will find a far easier way to use a number such as 25 than by multiplying by 2 and then 5 and adding. 77 is obviously 70 plus 1/10 of the product, while 63 is 70 minus 1/10 of the product. We can tackle 125 in several ways; for this use, we can consider it 100 plus ¼ of the product. 720 is 800 minus 1/10 the product.

In the choice of short-cut methods, and in the best use of each, you have great flexibility. There is no substitute for number sense here, for it is in finding the relationships that your key to method selection lies. There are so many variations, so many slightly different approaches, that it is up to you to select the fastest and easiest in each case.

Try these problems on your pad, finding an appropriate breakdown for each:

We have already covered the most convenient breakdown for 15. It was inserted here to remind you of the repetitive character of many useful breakdowns. In multiplying 895 by 10 and adding half the product, you would think simply "8950, plus 4475, is 13425."

In dealing with 473 x 38, you run into another fraction in your breakdown. 38 is 2 less than 40. 2, in turn, is 1/20 of 40. So, to multiply by 38, you can multiply by 40 and subtract 1/20 of the product, 1/20 is just ½ of 1/10. You do it like this:

Note two instructive points about this example:

First, you can (and should) jot down the answer to 40 times 473 from left to right without copying the original number. It is simply 4 x 473, digit by digit in the no-carry method, plus one zero.

Second, you can (and should) jot down from left to right the division of 18920 by 20 without any strain. You simply divide by 2 and start one place to the right when you put down the answer, which also divides automatically by 10. The combination results in a division by 20.

The third breakdown, 682 x 27, breaks down the 27 into 30 minus ¾o the product. Here again, you multiply 682 by 3 as you jot down the result and add one zero to make the multiplication by 30 instead of by 3. Under it you write the same digits one place to the right, without the zero, which automatically divides by 10, and then subtract:

Short cuts are a variety of methods, not a single system. There are many problems to which you can find short cuts, others to which you cannot, without doing more work than simplified arithmetic would involve. It comes down to recognizing the short cut that makes sense in a flash, because if you brood for more than an instant or two on whether or not to use a short cut at all, in that time your new systems of basic arithmetic could have finished most of the problem.

Try recognizing breakdown possibilities in these numbers:

50 45 24 33 54 63 82

Some of these begin to pioneer new breakdown possibilities that we have mentioned but not yet fully demon strated. Yet your own good number sense should show you interesting ways in each case.

50, for instance, is exactly half of 100. It is entirely up to you whether you find it easier and quicker to multiply by 5 and add a zero, or to add two zeros and divide by 2. Simple as it may seem, this is a perfectly valid short cut.

45 has been mentioned before, as 50 times the number, less 1/10 of the product. If you have not noticed it before, 45 is also 30 times the number, plus ½ the product. Which is better? Neither. It depends on the relationships of the numbers with which you are working, and on your own preferences.

24 is a new one. 24 is 20 times the number (double it and add a zero), plus 2/10 of the product. Jotting down 2/10 is simple: double the product, but start one place to the right. This divides by 10 and multiplies by 2 at the same time. If this seems at all obscure, follow the working in this example:

Usual way                                Breakdown way

This breakdown example emphasizes the value we put at the beginning of this chapter on being able to handle multiplication and division by 10, 100, etc., without hesitation or strain. This is the key to handling breakdowns such as the one above of doubling a number and then doubling the result —but multiplying by 10 in the first case and dividing by 10 in the second.

Our next number is 33. This is obviously 30, plus 1/10 of the product.

63 is based on the opposite principle. 63 can easily be broken down into 70, minus 1/10 of the product.

82 is a bit different. We will break 82 down into 80, plus 1/40 of the product. This is not difficult to handle. You simply divide the product by 4, but start writing your answer one extra place to the right. This divides by 4 and by 10 all at once —dividing by 40. See how it works:

Usual way                                Breakdown way

In a division such as the one above, you have an automatic running check on your accuracy because the division must come out even. If it does not, you know you have made a mistake. This is because two whole numbers, when multiplied, must give a whole-number answer. So if your division has a remainder, you are warned to recheck it.

There is no clear-cut advantage in this particular problem to breaking down 82 in the fashion we did, rather than into 80 plus twice the original number (which is merely a simpler expression of what our regular multiplication does). In one case you divide by 4 and 10; in the other you double. If the product of 80 x 5555 were part of the problem, however, it would be very tempting to divide the first product of 44440 by 4 and 10 to get the second line. Once again, which breakdown is best depends on how the numbers relate to each other.

By and large, the major value of breakdown is in permitting you to use easier-to-handle operations and digits. Breaking down 78 into 80 minus twice the other number, for instance, lets you substitute a simple doubling for a multiplication by 8 at the second step.

Breakdown—like any short-cut technique—is valuable to you only as you learn to handle it easily and well. Do not dismiss it out of hand if your first reading of a particular problem leaves you more baffled than enlightened, but on the other hand do not force yourself to use a particular short cut that after a few tries does not spring into your mind naturally and obviously. The purpose is to save work, not make it.

Longer Numbers

The easiest numbers to break down are usually those with two digits. But this does not mean that much longer numbers cannot also be broken down, frequently with dramatic results.

Take the multiplier 297, for instance. The nearest one-digit number that can form the base of your breakdown is 300. The difference between 297 and 300 happens to be a very convenient 1/100 of the product.
Note the same feature in the numbers 396—495—594. For each of them, you can substitute a multiplication by the next even hundred and subtract 1/100 of the product, instead of multiplying through by three digits and then adding all three lines.

In reverse, the same short cut is possible with 303— which you have probably used in the past without special instruction. There is no need to multiply twice by 3; merely copy the first product again, two places to the right, and add.

Now that you have learned to add or subtract 1/10 or 1/100 2/10 or 2/100, and so on, the possible range becomes considerably larger. You might handle 306, for instance, by doubling the first product two places to the right, rather than multiplying by 6.

As the breakdowns become more complex, so does the saving of time in using them. When you break down a three-digit number into two one-digit parts, you save a full digit in your work while at the same time performing a basically simpler operation.

For instance, consider the multiplier 784. There is no simple relationship between 700 and 84, so you do not break it down that way. 784 is just 16 less than 800, however, and 16 is exactly 2/100 of 800. So, instead of multiplying digit by digit by 784, we can multiply by 800 and subtract 2/100 of the product. Now the short cuts become visibly dramatic:

Usual way                                Breakdown way

In finding the first line of the working figures for the breakdown example, you do not copy the problem itself. You should be able to work without copying the problem with any single-digit multiplier if you have been working conscientiously on your no-carry multiplication. The second line of working figures, of course, is merely the first line doubled— two places to the right.

Since you are beginning to find it more and more natural to multiply by any single digit, you can extend your breakdowns into any number of tenths or hundredths. The number 558 might, at first sight, not show any exciting breakdown possibilities. 58 bears no reasonably simple proportion to 500. 558 is 42 less than 600. The key is to look at the 42 and the 6 in 600, and note—6 x 7 is 42. So you can multiply any number by 558 by first multiplying with 600 and then subtracting 7/100—which you do by multiplying the first product by 7 and putting down the answer two places to the right.

Choosing Multipliers

In all of our demonstrations so far, we have broken down the bottom number of the problem—the one normally considered to be the multiplier. Except when problems are set up for us in this fashion, there is of course no real "multiplier" and "number multiplied." In actual business or personal life, we simply need to multiply two numbers together, and it is up to us to decide which we will treat as the multiplier.

The reason this fact is worth special attention is that you can break down either number of a multiplication. As you start a particular problem, glance at both numbers for breakdown possibilities. The one you break down becomes your multiplier.

For example, you might face the problem 69 x 58. A quick look at 69 shows you that it can be broken down into 70, minus 1. 58 can be broken down, but not nearly as easily. So pick 69 as your multiplier.

Mixed through the various examples so far have been two different methods of breakdown. One is the special case called "rounding off and adjusting," in which you choose a convenient round number and then add or subtract the other number or a simple multiple of it to adjust. 69 is an example of this. So might be 68, since it is easier to multiply by 70 and then subtract twice the other number than it is to multiply it first by 6 and then by 8 and then add.

The second method is rounding off and adjusting by a fraction of the product of your first multiplication, rather than by the other number. For 63, you multiply by 70 and subtract 1/10 of the product. For 392, you multiply by 400 and subtract 2/10 of the product.

This difference should be crystal clear. In the first case, your difference is adjusted in terms of the number multiplied. In the second case, your difference is adjusted in terms of the product of your first multiplication.

Here, in order to make the difference very specific, is the same number broken down in each way:

Other-Number Adjustment First-Product Adjustment

48—50 minus 2 times   the other number
48—40 plus 2/10 the product
Which of the two breakdowns is better? Once again, neither. It depends on the other number and on the methods you yourself find easiest to handle. Either breakdown, you note, permits you to substitute a simple doubling for a multiplication by 8.

You can push the breakdown technique to impressive extremes. The nearest convenient one-digit multiplier may not be the next even ten or hundred at all; it may be two or more away. 1860, for instance, can be broken down so that you multiply by 60 and add 30 times the product. 328 can be-become: multiply by 8 and add 40 times the result.

This field of sophisticated breakdowns is fascinating, but it is too involved to be treated fully here. If you enjoy the idea, you can doodle for hours and find a breakdown for almost any number you may try. As genuinely useful short cuts, however, the more abstruse applications are questionable. You would spend more time breaking down your multiplier than the whole problem would take in simplified arithmetic. Number sense, again, is the real key. If you cannot "see" a relationship at one or two glances, then the short cut is not a real short cut for you.

The most useful ground rules for the two types of breakdowns are these:

ONE: If you round off one of the numbers to be multiplied, can you add or subtract the other number to adjust few enough times to be easier than the full multiplication?

TWO: If you round off one of the numbers to be multiplied, can you add or subtract a simple enough fraction of the first product to be easier than the full multiplication?

If the answer to either of these questions is yes, then breakdown can save you work and time in solving the problem. If the answer seems to be no, then another short cut may be in order.
 
Each of these numbers can be broken down in a way that before you check your reactions against the proposed breakdowns that follow.

Answering these two questions rapidly is the way to break down problems quickly and easily. See how many sensible breakdowns you can find in these multipliers:

In some cases, more than one breakdown is possible. We will give only the one that seems simplest and most generally useful. Since the breakdowns are of both types, we will use the shorthand N to mean that adjustment is in terms of the other number, and P to mean that adjustment is in terms of the product of the first multiplication.

For instance, our breakdown for the first number—58— is given as 60 — 2N. This means you multiply by 60 and then subtract the other number, doubled. The breakdown for 72 is given as 80 — 1/10P, which means you multiply the other number by 80 and then subtract 1/10 of the product.

Here are the breakdowns:

Two or three special notes are in order. The idea of "breaking down" 9 may seem peculiar. Yet it is possible, should you choose to use it; and you may well prefer to subtract a number from the same number (with an added 0) rather than to multiply through the entire number by 9.
 
The same comment applies to 90, of course. It is precisely the same breakdown, with one more 0 on both numbers.

Breaking down the number 26 into 20 plus 3/10 the product does not, in one sense, save any steps. The point here is that it offers you the choice of multiplying the other number by 6, or the first line of working figures by 3 (starting one place to the right). Other factors being equal, it is usually easier to multiply by the smaller of two digits—in this case, by 3 rather than by 6. So while breaking down 26 is not a short cut in the sense of saving steps, it does simplify the operation.

Breakdown in Subtraction

Ninety per cent of the value of breakdown is in multiplication. There is no easy way to use it in division, and it does not really save any time in addition. In subtraction, however, breakdown can sometimes speed up a problem if the relationship of the numbers is within a certain range.

The technique in subtraction is to raise the smaller number to the next-higher even number, then add the same amount to the larger number. This converts the problem into a form in which you can see the answer at a glance.

Suppose you need to subtract 64¢ from 98¢. Using the breakdown technique, you add 6 to 64 to make it an even 70. You adjust by adding 6 to 98 too, which then becomes $1.04. Subtracting 70 from 104 is a sight job. In subtracting 297 from 465, you add 3 to 297 to make it an even 300, and adjust by adding 3 to 465 to make 468. The answer, 168, is automatic.

The main application of this method is in adjusting numbers that fail by merely a digit or two of reaching the next even number. If the adjustment is much more than this, complement subtraction will be both easier and faster.

For such special cases, however, breakdown can be useful. Here is one example:

Usual way        Breakdown way

While you will not find such examples in your work every day, they do come up once in a while and this little trick is well worth keeping in mind.

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