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CHAPTER V

DECIMAL FRACTIONS

38. What Are Decimals?—In the old days, when no machinist pretended to work much closer than in. and the micrometer was unknown, the mechanic had little use for decimals except in figuring his pay. Now, however, we find that micrometer measurements are used so generally that a knowledge of decimal fractions is essential.

A Decimal Fraction is merely a fraction having a denominator of 10, 100, 1000, or some similar multiple of 10. The denominator is never written, however, but a system similar to that used in writing United States money is used. A decimal fraction is written by first putting down a period or "decimal point" and then writing the numerator of the fraction after the decimal point in such a manner that the denominator can be understood. Everything that comes after the decimal point (to the right of it) is a fraction, or part of a unit.

In writing sums of money, the first figure after the decimal point indicates dimes or tenths of a dollar; the second figure indicates cents, or hundredths of a dollar; the third figure, if any, indicates mills or thousandths of a dollar. This system has proved so convenient that it has been extended to represent fractions of any sort of a unit (not necessarily dollars).

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Let us take a decimal, say .253, and find out its meaning. We said that the first figure was tenths; the second, hundredths, the third, thousandths; and so on. Then .253 would be 180+100. This is not a very handy system unless there is

+

some easier way to read it. If we reduce these to a common denominator and add them, we get:

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This shows that one place to the right of the decimal point indicates a denominator of 10, two places a denominator of 100, three places a denominator of 1000, and so on. In other words, the denominator of a decimal fraction contains as many zeros as there are figures to the right of the decimal point in the decimal number. For example,

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.2749 is

2749 10,000

.042 is

42 1000'

or two thousand seven hundred forty-nine tenthousandths.

5' or forty-two one-thousandths.

10

5 1000.

The last case (.042) presents an interesting problem. Here we have a numerator so small in respect to the denominator that it is necessary to have a cipher, or zero (0) between it and the decimal point, in order that the denominator can be indicated correctly. Let us see how we would go about writing such a common fraction as a decimal. Take If we merely wrote .5 that would be 1% and would, therefore, not be right. From the rule for finding denominators of decimals we see that there must be as many figures after the decimal point as there are ciphers in the denominator. In this case the denominator (1000) has 3 ciphers, so we must have 3 figures in our decimal. We, therefore, put 2 ciphers to the left of the 5 and then put down the decimal point. We now have .005, which can be easily seen to be 1000.

One thing that must be carefully borne in mind is that adding ciphers after a decimal does not change the value of the fraction. .5 is the same in value as .50 or .500, because is the same in value as 100 or 1000. On the other hand, ciphers immediately

50

.500

0

following the decimal point do affect the value of the fraction, as has just been shown.

Mixed numbers are especially easy to handle by decimals, because the whole number and the fraction can be written out in a horizontal line with the decimal point between them. We read mixed decimals just as we would any mixed number-first the whole number, then the numerator, and lastly the denominator.

Example:

42,137.24697.

In this example, 42,137 is the whole number, and .24697 is the fraction. The number reads "forty-two thousand one hundred thirty-seven and twenty-four thousand six hundred ninety-seven hundred-thousandths.

The names and places to the right and left of the decimal point are as follows:

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When decimal fractions are written in words, the word "and" is used to imply the decimal point as indicated in the following: 2.5 is read "two and five-tenths."

652.47 is read "six hundred fifty-two and forty-seven hundredths."

42,123.9674 is read "forty-two thousand one hundred twentythree, and nine thousand six hundred seventy-four tenthousandths."

39. Addition and Subtraction. Knowing that all figures to the right of the decimal point are decimal parts of one thing and that all figures to the left are whole numbers and represent whole things, it will be seen readily that in addition and subtraction the figures must be so placed that the decimal points come under each other. As was shown under United States Money, the

operations can then be carried out just as if we were dealing with whole numbers.

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Pay no attention to the number of figures in the decimal. Place the decimal points in line vertically. You can, if you desire, add ciphers to make the number of decimal places equal in the two numbers. Remember, however, that the ciphers must be added to the right of the figures in the decimal. Proceed as in ordinary addition and subtraction, carrying the tens forward in addition, and borrowing, where necessary, in subtraction just as with whole numbers.

40. Multiplication. In multiplication forget all about the decimal point until the work is finished; multiply as usual with whole numbers. Then point off in the product as many decimal places, counting from the right, as there are decimal places in the multiplier and multiplicand together.

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Explanation: Since there are three decimal places in one number, and one in the other, we count off in the product four (3 + 1) places from the right and place the point between the 7 and the 4. The last 0 can be dropped after pointing off the product, giving the result 34.762 (or 34,782). The reason for this can be seen from the following: The whole numbers are 6 and 5. The result must be a little more than 6 X 5 30, and less than 7 X 6 42, since the numbers are more than 6 and 5, and less than 7 and 6. The actual result is 34.762.

=

=

The position of the decimal point can be reasoned out in this way for any example, but the quickest way is to point off from the right a number of decimal places equal to the sum of the numbers of decimal places in the multiplier and multiplicand.

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In the above examples it was necessary to put ciphers before the product in order to get the required number of decimal places. To see the reason for this take a simple example such as .2.3. The product is .06 or 18, as can be readily seen if they are multiplied as common fractions (%1% = 180). This checks with the rule of adding the number of decimal places in the two numbers to get the number in their product. The product of two proper fractions is always less than either of the fractions, because it is part of a part.

41. Short Cuts.-If we want to multiply or divide a decimal by 10, 100, 1000, or any similar number, the process is very simple. Suppose we had a decimal .145 and then moved the decimal point one place to the right and made it 1.45. The number would then be 15% or 148 instead of, so we see that moving the decimal point one place to the right has multiplied the original number by 10. Therefore, we see that:

45

100

To multiply by 10 move the decimal point one place to the right. To multiply by 100 move the decimal point two places to the right.

For other similar multipliers move the decimal point one place to the right for each cipher in the multiplier. This process is reversed in division, the rules being:

To divide by 10 move the decimal point one place to the left. To divide by 100 move the decimal point two places to the left, etc.

Example:

Reduce 10,275 cts. to dollars.

10,275 ÷ 100

=

$102.75 (decimal point moved two places to left). 42. Division. The division of decimals is just as easy as the multiplication of them after one learns to forget the decimal point entirely until the operation of dividing is finished. Divide as in simple numbers, disregarding the decimal point. Then to locate the decimal point, subtract the number of decimal places

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