Speed Mathematics Simplified (46 page)

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 3/8
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
of 100.
is of course the same as
, 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:

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 1/8. Since 125 is 1/8 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 5/8) 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 7/8 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: