Speed Mathematics Simplified (60 page)

The answer is 14 31/48. You got it by, first, translating 4 5/8 into 37/8. Then you translated 3 1/6 to 19/6. This multiplication shows no reduction possibilities, so you multiply top by top and bottom by bottom to get 703/48. Translate this back to a mixed number by dividing 703 by 48, and produce the final proper answer of 14 31/48.

Dividing by mixed numbers is just the reverse of multiplying. Translate each to an improper fraction, then
invert
the
divider
and multiply as always.

Let's do the last example as a division:

The two mixed numbers translate to the same improper fractions: 37/8 and 19/6. Since this is division, however, we turn the divider upside down and handle it as a multiplication:

Pause to look for reduction possibilities. The 6 and the 8 are both divisible by 2, so we can simplify the problem a bit to read:

Multiplying top by top and bottom by bottom, we get the answer 111/76. This is an improper fraction, but the top is not twice the bottom so we do not divide. We put down a 1 for the whole-number part, and subtract the bottom from the top to find the top of the fractional part. The final answer is 1 35/76.

Most of the fractions and examples in this chapter have been more complex than the ones you normally run into in your number work. This has been entirely on purpose. Learn to handle those in this chapter well, and simpler ones should be easy.

19

SPEED AND EASE IN DECIMALS

F
RACTIONS are one way of expressing quantities of less than 1, or more than 1 but not reaching an exact digit. Decimals are another way of doing the same thing.

Of the two, decimals are usually by far the easier and more convenient way to express fractional quantities. If our measuring systems were based on our ten-base counting system (as is the Continental system of meters, grams, litres, and so on) we would perhaps face fractions only a very few times in our lives. But since we have inherited a jumbled group of weights and measures broken down variously into twelfths (feet), sixteenths (pounds), sixtieths (hours), fourths (gallons) and even 5,280th's (miles), we face fractions all the time.

Only in our U. S. money system are we blessed with a commonsense decimal progression. In all our other measurements, we cling to outrageous counting bases.

Even for these, however, decimal fractions are usually accurate enough. They are not as perfect an expression of many quantities as are fractions, which can express any conceivable quantity with exact preciseness, but the difference is so slight as to be meaningless in most cases. In fact, many of the quantities we consider “hard” or “exact” are only approximations to begin with. 5 apples is precisely 5 apples, but
5 inches or 5 pounds is only as (approximately) close to 5 inches or 5 pounds as our measuring equipment can determine at the time.

Actually, these are two completely different types of numbers. One is a precise quantity; the other is a declaration of comparison to an artificial standard such as acres or gallons. Think for a bit about the essential differentness of the two approaches to numbers, for the sake of your number sense.

As far as preciseness of decimals goes, 1/3 is a prime example. There is no decimal equivalent, nor can there ever be. The fraction 1/3 expresses a certain quantity with complete accuracy. The decimal 0.3333333333333333333 approaches 1/3, but it is
not
1/3. No matter how many 3's you add, you never quite reach 1/3. .33 is accurate to 1 part in 100, however, while .333 is accurate to 1 part in 1,000. For most practical needs, this is more than enough accuracy.

A decimal is a shorthand way of expressing a fraction that has a bottom of 10, 100, 1,000, or some other multiple of 10. We use the decimal point, the little period, to indicate that the digits following it do not express a whole-number quantity, but a fraction whose bottom is a multiple of 10. The number .3 is the same as
. 1.3 is the same as l,
. The point tells us when to stop figuring in whole numbers and begin noting the fraction.

The first lesson usually taught in reference to decimals is how to read them properly. Do you read 0.33 as 33 tenths, hundredths, or thousandths? There is a beautifully simple and reliable trick that removes any possible confusion. Merely pretend that the decimal point itself is a 1, followed by as many 0's as there are digits after the point. This imaginary number is the bottom of your fraction.

Thus 0.3 must be
, since the “1” (point) is followed by one digit, and 10 is ten. 0.33 must be
, because there are two digits after the point and two 0's in 100.

See if you can read 0.4567 with this method. The top of the fraction is 4567, of course. The bottom is a 1 followed by four 0's. So the bottom must be 10,000.

If there are any zeros immediately following the point,
count them as digits too in figuring the bottom. 0.03 is
, not
, because there are two digits after the point.