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 1, as can be readily seen i they are multiplied as common fractions (X=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. 39. 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 145 instead of 15; so we see that moving the decimal point one place to the right has multiplied the original number by 10. Therefore, we see that: 4 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 10275 cents to dollars. 10275÷100-$102.75 (Decimal point moved two places to left). 40. 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. Then point off from the right as many decimal places in the quotient as the number of decimal places in the dividend exceeds that in the divisor. In other words, we subtract the number of decimal places in the divisor from the number in the dividend and point this number off from the right in the quotient. It makes no difference if the divisor is larger than the dividend, as in the following example. In such a case the quotient will be entirely a decimal. Example: 22.76284.25=? 84.25)22.762000(.2701+ or .2702, Answer. 16 850 5 9120 5 8975 14500 6075 Explanation: The divisor being larger than the dividend, the quotient turns out to be an entire decimal. In this case we will presume that we wanted the answer to four decimal places. We have, therefore, added ciphers to the dividend until we have six decimal places. When these have all been used in the division, we have 6-2-4 places in the quotient. The remainder is more than half of the divisor, showing that if we had carried the division to another place, the next figure would have been more than 5. We, therefore, raise the last figure (1) of the quotient to 2, because this is nearer the exact quantity. In stopping any division this way, if the next figure of the quotient would be less than 5, let the quotient stand as it is, but, if the next figure would be 5 or more, as in the example just worked, raise the last figure of the quotient to the next higher figure. Sometimes the decimal places are equal in dividend and divisor, as for instance, if we divide .28 by .07. .07).28 As the numbers of decimal places in the dividend and divisor are the same, the difference between them is zero, and there are no decimal places in the quotient. The answer is simply 4. The decimal point would come after the 4 where it would, of course, be useless. If there are more decimal places in the divisor than in the dividend, add ciphers at the right of the decimal part of the dividend as far as necessary. In counting the decimal places, be sure to count only the ciphers actually used. 41. Reducing Common Fractions to Decimals.-Common fractions are easily reduced to decimals by dividing the numerator by the denominator. In the case of, we divide 1.0 by 2 and get .5. All that is necessary is to take the numerator and place a decimal point after it, adding as many ciphers to the right of the decimal point as are likely to be needed, four being a common number to add, as four decimal places (ten thousandths) are accurate enough for almost any calculations. If is to be reduced to a decimal, the work is simply an example in long division, the placing of the point being the main thing to consider. Simply divide 1.00000 by 32. This gives .03125 or 3125 one hundred-thousandths. 32) 1.00000(.03125 96 40 32 80 64 160 42. Complex Decimals.-A complex decimal is a decimal with a common fraction after it, such as .12, .0312, etc. The fraction is not counted in determining the number of places in the decimal. .12 is read "twelve and one-half hundredths." .0312 is read "three hundred twelve and one-half ten-thousandths." To change a complex decimal to a straight decimal, reduce the common fraction to a decimal and write it directly after the other decimal, leaving out any decimal point between them. 43. The Micrometer.-The micrometer is a device to measure to the thousandth of an inch and is best known to shop men in the form of the micrometer caliper shown in Fig. 6. The whole principle of the micrometer, as generally made, can be said to depend on the fact that of 100. The micrometer, as shown in Fig. 6, is made up of the frame or yoke b; the anvil c, the screw or spindle a, the barrel d, and the thimble e. The spindle a is threaded inside of d. The thimble e is attached to the end of the spindle a. The piece to be measured is inserted between c and a, and the caliper closed on it by screwing a against it. The screw on a has 40 threads to the inch, so if it is open one turn, it is open in., or 56, or .025. Along the barrel d are marks to indicate the number of turns or the number of fortieths inch that the caliper is open. Four of these divisions () will represent one-tenth of an inch, so the tenths of an inch are marked by marking every fourth division on the barrel. Around the thimble e are 25 equal divisions to indicate parts of a One of these divisions on e will, therefore, indicate turn. of a To read a micrometer, first set down the number of tenths inch as shown by the last number exposed on the barrel. Count the number of small divisions on the barrel which are exposed between this point and the edge of the barrel. Multiply this number by .025 and add to the number of tenths. Then observe how far the thimble has been turned from the zero point on its edge. Write this number as thousandths of an inch and add to the reading already obtained. The result is the reading in thousandths of an inch. Example: .7 .025 .018 Let us read the micrometer shown in Fig. 6. .743 in., Answer. Explanation: First we find the figure 7 exposed on the barrel, indicating that we have over in. This we put down as a decimal. In addition, there is one of the smaller divisions uncovered. This is .025 in more. And on the thimble, we find that it is 3 divisions beyond the 15 mark toward the 20 mark. This would be 18, and indicates .018 in. more. Adding the three, .7+.025+.018-.743 in., Answer. This can perhaps be better understood as being 7 thousandths less than in. Lots of men locate a decimal in their minds by its being just so far from some common fraction. Most micrometers have stamped in the frame the decimal equivalents of the common fractions of an inch by sixty-fourths from in. to 1 in. A table of these decimal equivalents is given in this chapter, and will be found very useful. Everyone should know by heart the decimal equivalents of the eighths, quarters, and one-half, or, at least, that one-eighth is .125. Then =5X.125=.625; and }=7×.125=.875, etc. Also, if possible, learn that.062, or .0625. To get the decimal equivalent of a number of sixteenths, add .062 to the decimal equivalent of the eighths next below the desired sixteenths. Example: What is the decimal equivalent of 13 3 1 = 1 3121⁄2 16-1+16-.750+.062.812 in., or .8125 in. To set a micrometer to a certain decimal, first unscrew the thimble until the number is uncovered on the barrel corresponding to the number of tenths in the decimal. Divide the remainder by ,025. The quotient will be the additional number of the |