Chapter 17 - Choosing And Combining Short Cuts

You have learned and practiced the four most generally A useful short cuts. There are others, but they are quite specialized. The most complete assortment can be found in the books listed in the bibliography. With the four short cuts you have learned, however, you can convert a great deal of your multiplying and dividing into simpler forms.

Review all together in one place the four different approaches:

BREAKDOWN For one of the numbers to be multiplied, use a round number if this permits an adjustment with an easy fraction of the other number or of the result of the first step. 39 becomes 40 - 1; 45 becomes 50 - 1/10 the first product.

ALIQUOTS When one of the numbers is an even fraction of a ten-base, use the fraction instead of the number. 25 is treated as ¼ of 100.

FACTORS When one of the numbers can be factored, multiply or divide by each of the factors in turn. 63 is treated as 9, then 6.

PROPORTIONATE CHANGE When one of the numbers can be simplified by doubling or halving it (or any other such change), use the simpler form and compensate the other number or the answer. 35 becomes 70, with a compensating factor of 2.

Many numbers can be short-cut with not just one, but with two or more of these methods. 45, for instance, can be factored (5 x 9), broken down (50 less 1/10 product), or changed (90, compensate with 2). An interesting exercise is to locate one number to which all of these methods can apply. One such number is 125. Witness the various short-cut handlings of the number 125:

BREAKDOWN: 100, plus ¼ product.

ALIQUOTS: ⅛ of 1,000.

FACTORS: 5x5x5.
PROPORTIONATE CHANGE: quadruple to 500.

For real mastery of short cuts, try to get a feel for the real identity of these four apparently different relationships. One of the techniques will work out in even-number terms for a number that none of the others might handle in this way, but essentially they are all merely different expressions of the same fundamental situation.

The aliquot approach to 125, for instance, is to take ⅛ of 1,000. The proportionate change approach is to use 500, and compensate by a factor of 4. 500 is just half of 1,000, and 4 is just half of 8. The relationships are the same; only the facets we choose to see in any one case appear to be different.

Numbers to which all four approaches apply without remainders or fractions are few, but numbers for which two of the short cuts work are plentiful. Exercise your understanding of the four approaches by converting each of the following numbers in at least two ways:

75        72        36 4½  63        384

Cover the explanations with your pad until you have found two or more short cuts for each of these numbers. Here are the possibilities:

This does not mean that the different short cuts are of equal value whenever a number can be converted in two or more different ways. The value in each case depends not only on the possibilities of that number, but also on the other number involved. It also depends on which of the ways you find most adapted to your own ease and speed. The idea is to pick the simplest method for working the problem. Sometimes it will be one of the short cuts, and other times it will be to go ahead and do it with your new streamlined arithmetic. Flexibility is the key. You do not dig a hole for a rose bush with a steam shovel, or use a garden spade for a house foundation. Equally, you do not use three two-digit factors in place of multiplying by the number you factored, because it would not save you any work.

Combining Short Cuts

The possibilities of combining two short cuts in one problem are quite extensive and rather intriguing.

Breakdown by itself, for instance, makes sense only if you can break a number down to a simple base and an adjustment that is a simple fraction of the other number or the product. If you combine methods, however, you can use any aliquot or any easily factored or any easily changed number as a base.

In the range of numbers in which breakdown alone would save you work, 25 was not a useful base because you would still have to multiply through by two digits. But with aliquots to help, you could break down 26 into the combination technique "divide by 4 (aliquot) plus the other number (breakdown)."
 
Here is how you would do it:

Usual way                                Aliquot-breakdown way

Note that since you are dividing by the easy-to-handle divider 4, your short-cut method is to jot down only the answer.

Surprisingly difficult-looking problems can sometimes be solved almost at sight when one of the numbers happens to contain an aliquot as an easy breakdown base. 1375 x 8642 becomes merely ⅛ of 86420000, plus 1/10 of the product— because 1375 can be broken down into 1250 (⅛ of 10,000) plus 125 (1/10 of 1250). Try it and see.

Do the following problem with an aliquot-breakdown.

Cover the explanation with your pad while you give it your best.

We break the number 385 into 375 plus 10—⅜ of 1,000 plus 10 times the other number. First you divide 4782000 by 8, jotting down only the answer. Multiply the result by 3. Then add 47820—10 x 4782:

Just for comparison, here is the usual way of solving the same problem:

Try breaking down these multipliers to aliquot bases:

Some of these get a little tricky, but each of them can be broken down to an aliquot base. Here is how:

Just as you can break down a complex multiplier to an aliquot base as well as a rounded-off base, so can you break down multipliers to a factorable base. There would be no point in breaking down 37 by the breakdown method alone. But by combining it with the factor short cut, 37 becomes 6X6 (factors) plus the other number (breakdown).

See if you can find the surprisingly easy breakdown-factor short cut for this problem:

Any series of digits that repeats itself—such as 81, 81 —is a very automatic breakdown. This number is 8100 plus 1/100 of itself. 8100, in turn, is at sight 90 x 90. So the breakdown-factor short cut would be: "90 x 90 (factors) plus 1/100 of the product (breakdown)." In addition, however, we often handle multiplication by 9 as a breakdown, using 10 — 1. In this case, 90 is 100 - 10. Let us show all three methods of working:

Usual way

As very often happens, the illustrations of the two short cuts do not dramatize the real simplification involved: the handling of easier processes at each step. Follow each of them pencil in hand to see how this works.

Breakdown can also sometimes be combined with proportionate change. The number 34 does not find a natural place in any one of the individual short-cut methods. But if you realize that 34 is just one less than an easy proportionate-change base, you might choose to handle it as a multiplier as 70 (proportionate change) and cut the other number or answer in half; then subtract the other number (breakdown).

You might or might not choose to use any of these specific combinations. Again, and again: hunt for relationships, use the short cut or combination of short cuts that flashes into your mind as an easy and sensible method, and get the problem done. This, after all, is the end purpose of all mathematics, short-cut or not; get the problem done.

Other Combinations

The possible combinations of short cuts are almost endless. A book could be written about the refinements of double and triple and quadruple combinations of methods. It would be an interesting exercise, but would not really get you through your arithmetic with greater speed and accuracy except for the particular relationships that happen to hit you with special and memorable force.
One or two other wrinkles would, however, speed up your number work from time to time. They are rather intriguing, too.

We noted a page or two back that sometimes you will break a multiplier down to a factorable base. You will, as well, discover sometimes after you have factored a number that one of the factors is too complex to save much time in using the straight factor approach—but that complex factor might be broken down. This is just the reverse of breakdown-factor; it is, if you will, factor-breakdown.

Let us try one. As a start, factor the multiplier 261.

The digit sum of 261 is 0, so you know 9 is a factor. 9 into 261 gives you 29 as the other factor.

Now 29 is a prime number and it is a two-digit number, so factors do not short-cut this problem as much as we should like. But 29 is a very natural candidate for breakdown. We might solve a multiplication involving 261 by multiplying by 9, then 30, then subtracting.

Be very careful here to subtract, not the other number, but the product of 9 times the other number. Why? Because that 30 — 1 is a factor, not a breakdown of the whole number. Work the factors backward, if you wish, to get this point clear. 9 x 30 is 270. Subtract 9 (not 1) from 270 to get 261, the number we started with.

Here is an example involving this specific factor-breakdown:

Usual way                                Factor-breakdown way

We have covered only the combinations involving breakdown because they are the most generally useful. Some combinations do not make any sense at all, such as factoring to an aliquot. Play with the idea on your pad and you will see why.

The ultimate short cut is to have so firm a grip on your number sense and on the possible short cuts—together with useful combinations of them—that in each case you can quickly and unerringly pick the shortest, easiest road to the solution.

The next step is obvious. It is to practice, on some actual examples, the best approach to each. Do not bother to solve the following problems unless you wish to. The exercise is simply to select, in each case, the best technique. Keep in mind as you go through the exercise that in multiplication you might choose to short-cut either number, not just the one that appears on the bottom.

Examine each of the following problems for all reasonable short-cut possibilities and definitely state to yourself how you would tackle it before going on to the suggested approaches. Not all of them, by the way, should be converted. In four cases, there is no short cut possible. For practice, however, spend more time with each than you would expect to spend looking for short cuts in your work with figures.

Don't skip over the above exercise. Short of knowing the short cuts themselves, it is the most important practice in the short-cut section of this book. It does little good to know several short methods if you cannot see quickly whether or not each can be used.

In some of the above problems more than one conversion can be applied. You can treat the divider 25 in problem 18, for instance, as 5 x 5, or ½ of 50, or ¼ of 100. The suggested short cuts below, however, are those I believe simplest in each case. You are perfectly free to choose a different one if it will work and if it is easier for you.

1. Convert the top number into ⅛ of 10,000.

2. No practical short cuts. Do it straight.

3. All short cuts are not complicated. It is still easier to multiply by 9 by subtracting the number from 10 times the number.

4. You can factor 35 into 7 and 5, or double it to 70.

5. Two-step short cut. 126 is 125 plus 1, and 125 is ⅛ of 1,000.

6. Here is a reverse aliquot. Far simpler to multiply by .8 than by 4/5.

7. 47 is 1 less than 48, which is 6 x 8.

8. Treat 79 as 80 - 1.

9. Reverse this fraction to its aliquot form: .4.

10. The most elementary of all short cuts. 99 is 100 ―1.

11. Factor 378 into 9, 7, and 6. You divide three times, but by a single digit each time.

12. 69 is, of course, 70-1.

13. Factor the 72 into 9 and 8.

14. Choose among factoring the 45 into 9 and 5; doubling it to 90; or breaking it down to 50 - 1/10 product.

15. 75 is ¾ of 100. Instead of multiplying by 75, just multiply by 3(00) and divide by 4.

16 1 would convert the top number on this, although 97 is easy as 100 - 3. But 180 is twice 90, so subtract 10 97's from 100 97's and double the answer.

17. 625 is an aliquot, being ⅝ of 1,000.

18. Don't ever divide by 25. Subtract two zeros from the number divided (using a decimal) and multiply by 4.

19. 375 is ⅜of 1,000. Subtract three zeros; multiply by 8; then divide by 3.
 
20. The digit sum of 432 is 0, so you can factor it: 9, 8, and 6.

21. You cannot do much to the bottom number, but 5 is obviously a factor of the top number. A quick sight-division shows that the other factor is 49,
which in turn you factor to 7 and 7.

22. No practical short cut.

23. You should recognize factorable numbers of 81 or less at a glance. 49 is 7 x 7.

24. Do not let the decimal fool you. 4.5 can be handled just like the 45 in problem 14—but keep track of the decimal point.

25. No short cut would be worthwhile here.

26 Factor the top number in this problem. 256 is the product of 8, 8, and 4.

27. This is the last of the booby traps. Use your noarry, left-to-right multiplication for quick results.

28. This divider can be converted to a single digit with proportionate change. Divide by 900 and multiply by 4.

29. 875 is a perfectly good aliquot. Subtract 4 zeros, then divide by 8 and multiply by 7.

30.       63 is an easy breakdown: 70 — 1/10 product.

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