Speed Mathematics Simplified (44 page)

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
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:

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.

14

ALIQUOTS

T
HIS fascinating and useful technique of conversion suffers under a traditional and foreign-sounding name. “Aliquot” means, simply, an exact fraction. The word is derived from a Latin word meaning some, or several. It is usually used as an adjective (aliquot parts, meaning exact parts), but since it is also a noun we will save words.

The key word in the definition is
exact.
8 is an aliquot of 16, because it is contained within 16 exactly twice and leaves no remainder.

Since we count by the decimal system, based on ten, the aliquots of most use to us in short-cut mathematics are aliquots of ten, a hundred, a thousand, and so on. Incidentally, the word is pronounced ali-kwut.

We all think of 25¢ or “a quarter” as completely interchangeable, without giving it a second thought. We have dealt in quarter-dollars so much that we know by instinct that 25¢ is one quarter of 100¢. The special usefulness of this and many other aliquots (for 25 is indeed an aliquot of 100) may or may not have been brought to your attention.

For instance, you can multiply by 25 by adding two zeros to the other number and then dividing by 4:

The value of aliquots is not restricted to the number 25 (or its equivalents 250, 2500, 2.5, .25, and so on). Half of 25 is 12½, and 12½ is a number we meet surprisingly often. It is exactly 1/8 of 100. The same aliquot shows up as 125 (1/8 of 1000), as 1.25 (1/8 of 10), as .125 (1/8 of 1).

You might soon need to multiply 965 by 12.5. Which of these two ways looks easier?

The number 5 is also an aliquot, of course. It may be a tossup whether you would prefer to multiply by 5, or add a 0 and divide by 2. It depends on which you find easier. A very similar approach was suggested for 50 in the chapter on breakdown, incidentally; this illustrates the overlapping nature of some of the features of the different short-cut methods.

There are only 11 useful exact aliquots in the decimal system, but they are number combinations that show up very often. In addition, there are a number of
approximate
aliquots which can prove useful in estimating—such as 33 for 1/3 of 100—but be sure to remember that they are not real aliquots at all.

Here are the 11 aliquots. In order to avoid decimals, we will show them as aliquots of 1,000. Adding zeros, or moving decimal points to the left, can make these same numbers
prove to be aliquots of anything from 1 to any number of million you wish.

Exact Aliquots

All the 16th's, by the way, are exact four-digit aliquots, except
, but since the fraction is in two digits (16) their utility for short-cut arithmetic becomes somewhat remote.
of 10,000 is precisely 625, while
of 100,000 is 1875.
—naturally—is the same as 1/8, which appears in the table above.

Even aliquots with top and bottom digits (such as 3/8) can save work, because the number 375 for which 3/8 is the aliquot contains three digits. In order to multiply by 375 in the aliquot way, you first divide by 8 (after adding three 0's to the other number, since 375 is 3/8 of 1000) and then multiply the result by 3. Although you first divide and then multiply, this is still a little simpler than multiplying through with each of three digits and then adding the three lines of partial products.