Speed Mathematics Simplified (45 page)

Here is a comparison of the two methods:

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

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.

625 is an aliquot of 1,000, being 5/8 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:

The second-stage aliquot solution here can come in multiplying the 456,750 × 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 (7/8) 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
does not have a precise decimal equivalent. In other terms,
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
. 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.

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.