Chapter 14 - Aliquots

This 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 l00¢. 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:

Usual way        Aliquot way

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 ⅛ of 100. The same aliquot shows up as 125 (⅛ of 1000), as 1.25 (⅛ of 10), as .125 (⅛ 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 ½ 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 1/16, but since the fraction is in two digits (16) their utility for short-cut arithmetic becomes somewhat remote. 1/16 of 10,000 is precisely 625, while 3/16 of 100,000 is 1875. 2/16—naturally—is the same as ⅛, which appears in the table above.

Even aliquots with top and bottom digits (such as ⅜) can save work, because the number 375 for which ⅜ 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 ⅜ 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.

Here is a comparison of the two methods: Usual way
 
Usual way

Aliquot way

Do one on your own now. Cover the explanation that follows with your pad until you have solved this problem with an aliquot:

This is a very simple one, but you may be surprised at how much work an aliquot can save you even in a case like this.

Multiplying 24747 by 25 is, naturally, precisely the same as dividing 100 times 24747 by 4. So that is what we do. Our answer is 625 is an aliquot of 1,000, being ⅝ of it. Instead of multiplying by 625, then, we can divide 1,000 times 2654 by 8 and then multiply the result by 5. First, let us show the straight comparison:

Work out the answer to the problem in the traditional way and look at the two workings, side by side. The difference is quite dramatic.

Try one more, before moving on to other applications of the aliquot short cut. The following problem can be solved by using two aliquots, one for each stage of the solution. See if you can decipher this:

As always, cover the explanation with your pad until you have finished.

Usual Way

The second-stage aliquot solution here can come in multiplying the 456,750 X 5. If you find it easier to add a 0 and divide by 2 instead of multiplying by 5, you can easily set up this step into the answer of the first. Your working then looks like this:

Even in so complex a solution as this, the aliquot method obviously involves fewer working figures. Compare it with the standard solution once more.

Special Aliquots

The fact that many of our measuring systems are non-decimal (not based on ten) gives them different sets of aliquots. ¼ of ten, for instance, is 0.25. But the gallon is based on eight, not ten (two pints in a quart, four quarts to a gallon), so in terms of pints ¼ of a gallon is 2.

This gives us an occasional and interesting interplay between regular ten-base aliquots and gallons, feet, yards, hours, and other non-decimal measurements.
 
We can see at a glance that one pint is precisely 0.125 gallon. If we need to know how many pints are in 0.8750 gallon, we find that the 8 in the fraction form of the aliquot 875 (⅞) is wiped out by the conversion from decimal to pints-gallons, and we are left with an even 7 pints.

Inches to feet is a little tougher, since 1/12 does not have a precise decimal equivalent. In other terms, 1/12 is not an aliquot of the ten-base system, because its decimal equivalent is .0833+, with 3's going on forever because it never becomes exact. It is very close, however, so except for complete scientific accuracy you will find it accurate enough.

To find the number of inches in 0.9166 feet, then, you would note that the approximate fraction of .9166 is 11/12. In converting from decimal to duo-decimals (dozens), the 12 gets dropped and you have 11 inches.

Here are the most frequently used approximate aliquots. Remember that these are not true aliquots, because they are not precise, but they are close enough for a great deal of your number work.

Approximate Aliquots

based on 1,000

It is interesting to note that all the approximate aliquots are based on thirds and multiples of thirds—sixths and twelfths. This is inherent in the ten-based (decimal) system.

An extra bonus in the use of aliquots to bridge the difference between a ten-base system and an eight-, twelve-, or other-base system in weights and measures is that becoming aware of the aliquot equivalents is one of the best exercises you can give your number sense.

Try it once yourself. Using aliquots, figure out the number of pints in 375 gallons.

It should not take long. 375 is an exact aliquot, being ⅜ of 1,000. Since there are 8 pints in a gallon, there would be 8,000 pints in 1,000 gallons. The 8's cancel out, and you are left with 3,000 pints.

How many months in 83 years, for a quick guess? 83 is an approximate aliquot, about 5/6 of 100. 5/6 is of course the same as 10/12, and there are 12 months in a year. So the 12's cancel out, and we have about 10 times 100—or a thousand months. Actually it is 996, so we are .4 of 1% off.

Dividing with Aliquots

Unlike breakdowns, aliquots are just as valuable in dividing as in multiplying. When you divide with an aliquot, you simply reverse the rule for multiplying.

In multiplying, you multiply by the fractional form of your aliquot. In dividing, you divide by the fraction.

In multiplying, you add enough zeros to the other number to make the aliquot stay in proportion. 50 is ½—of 100 —so to multiply by 50 with a division of 2, you first add two zeros to the other number.

In dividing, you subtract as many zeros as you need to. Usually, you must use a decimal point.

Let us start with one of the simpler aliquots. Here is how you use the aliquot 25 for dividing:

Usual way        Aliquot way

Note that we subtracted two zeros from the number divided by using the decimal point. We subtracted two zeros because 25 is ¼ of 100. If we had been dividing by 2.5, we would have subtracted one zero because 2.5 is ¼ of 10. Dividing by 250 would require us to subtract three zeros.

The reason you almost always have to use a decimal point to subtract zeros when dividing with an aliquot is that division often does not come out even. The example above was a simplified introduction. If the number to be divided were 9643, of course, then we know simply by inspection that there would be a remainder because subtracting two zeros (by moving the decimal to the left) from 9643 gives us 96.43, and those two digits to the right of the decimal must be multiplied too.

Try a longer division yourself. Cover the answer with your pad until you have finished:

The proper aliquot form to use for 125 is ⅛. Since 125 is ⅛ of 1,000, we subtract three zeros from the number divided. Here is how the problem is set up:

If you feel ambitious, you might try dividing 73984 by 125 in the usual way to see if you get the same answer—and to compare the amount of work involved.

Since you multiply from left to right, you may not have to finish this multiplication all the way through. Carry it to the accuracy you need and then stop. If you need only the nearest tenth, work it out through the 7 and round off your answer to 591.9.

In aliquots with two digits, you again reverse the multiplication process. In multiplying, you divide by the bottom figure of the fraction (the 8 in ⅝) and then multiply by the top digit. In dividing, you multiply by the bottom digit and then divide by the top. This, naturally, is equivalent to division by the fraction.

Suppose we go through the following problem with an aliquot solution:

First, determine the aliquot. 87.5 is ⅞ of 100. Since we are using a fraction of 100, we subtract two zeros from the other number and start by multiplying it with the bottom of the fraction:

Now—and you would not bother to rewrite the result in actual practice—you divide by the top of the fraction:

This is obviously much easier than dividing, even in shorthand long division, by a three-digit number.

Turn to a clean page of your work pad now and tackle this problem with an aliquot solution:

Cover the answer with the pad.

The fraction for the aliquot 75 is ¾ of 100. First we subtract two zeros (we can simply omit them here, since there are two zeros) and multiply by the bottom of the fraction:

Now divide this product by the top of the fraction. In practice you would do it at sight:

That is all there is to it. Without dividing by anything more difficult than the single digit 3, you know that 29700 divided by 75 is 396.

Reversing Aliquots

If you will turn back for a moment to the table of exact aliquots, you will note that several of them are really simpler in their decimal form than they are in their fractional form.
 
A later chapter will cover fractions and decimals. If this special application of their interchangeability in terms of aliquots is at all confusing, it might be a good idea to refresh your memory with that chapter first.

The fraction 4/5, for instance, has the aliquot form .8. The decimal form of 2/5 is .4.

This fact makes possible a reverse short cut whenever you must deal in fractions that are more simply expressed in decimals. Rather than suffer through the fraction, use the simpler form.

One example should illustrate this sufficiently. Consider this problem:

This problem would traditionally be solved by multiplying 3 x 287 and then dividing the product by 5. But it is far, far easier to multiply 287 x .6:

The important lesson in this reverse-aliquot approach is that no single method is always best. The point is to learn awareness of the many different ways of accomplishing the same result, and to be on the lookout for the easiest and quickest in each particular case.

Sometimes it will be breakdown. Sometimes it will be the use of an aliquot. And sometimes it will be the use of factors—quite a different short cut.

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