Speed Mathematics Simplified (54 page)

Any series of digits that repeats itself—such as 81, 81—is a very automatic breakdown. This number is 8100 plus
of itself. 8100, in turn, is at sight 90 × 90. So the breakdown-factor short cut would be: “90 × 90 (factors) plus
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:

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 × 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-break-down:

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 × 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.	Â	Convert the top number into 1/8 of 10,000.
  2.	Â	No practical short cuts. Do it straight.
  3.	Â	All short cuts are not complicated. It is still easier to multiply by 9 by subtracting the number from 10 times the number.
  4.	Â	You can factor 35 into 7 and 5, or double it to 70.
  5.	Â	Two-step short cut. 126 is 125 plus 1, and 125 is 1/8 of 1,000.
  6.	Â	Here is a reverse aliquot. Far simpler to multiply by .8 than by .
  7.	Â	47 is 1 less than 48, which is 6 × 8.
  8.	Â	Treat 79 as 80 – 1.
  9.	Â	Reverse this fraction to its aliquot form: .4.
10.	Â	The most elementary of all short cuts. 99 is 100 – 1.
11.	Â	Factor 378 into 9, 7, and 6. You divide three times, but by a single digit each time.
12.	Â	69 is, of course, 70 – 1.
13.	Â	Factor the 72 into 9 and 8.
14.	Â	Choose among factoring the 45 into 9 and 5; doubling it to 90; or breaking it down to 50 – product.
15.	Â	75 is ¾ of 100. Instead of multiplying by 75, just multiply by 3(00) and divide by 4.
16.	Â	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.	Â	625 is an aliquot, being 5/8 of 1,000.
18.	Â	Don't ever divide by 25. Subtract two zeros from the number divided (using a decimal) and multiply by 4.
19.	Â	375 is 3/8 of 1,000. Subtract three zeros; multiply by 8; then divide by 3.
20.	Â	The digit sum of 432 is 0, so you can factor it: 9, 8, and 6.
21.	Â	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.	Â	No practical short cut.
23.	Â	You should recognize factorable numbers of 81 or less at a glance. 49 is 7 × 7.
24.	Â	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.	Â	No short cut would be worthwhile here.
26.	Â	Factor the top number in this problem. 256 is the product of 8, 8, and 4.
27.	Â	This is the last of the booby traps. Use your nocarry, left-to-right multiplication for quick results.
28.	Â	This divider can be converted to a single digit with proportionate change. Divide by 900 and multiply by 4.
29.	Â	875 is a perfectly good aliquot. Subtract 4 zeros, then divide by 8 and multiply by 7.
30.	Â	63 is an easy breakdown: 70 – product.