in the divisor from the number of decimal places in the dividend. The difference is the number of decimal places in the quotient. If there are more decimal places in the divisor than in the dividend, as, for example, .0064.00008, then a few ciphers can be added to the right of the dividend, making it .006400 ÷ .00008. This does not change the value of the dividend, and makes it possible to perform the division and locate the decimal point as before. When counting the decimal places, be sure to count only the ciphers actually used in the division. It makes no difference if the divisor is larger than the dividend, for in such a case the quotient will be entirely a decimal. Example: 22.762 84.25 = ? 84.25)22.762000(.2701 + or .2702, Answer. 16 850 5 9120 5 8975 14500 8425 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 4 decimal places. We have, therefore, added ciphers to the dividend until we have 6 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 ending 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, increase 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. 43. 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, 4 being a common number to add, as 4 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 44. Complex Decimals.-A complex decimal is a decimal with a common fraction after it, such as .12, .03121⁄2, 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 de b 0 1 2 3 4 5 6 7 20 a e 15 FIG. 9.-Micrometer caliper. common fraction to a decimal and write it directly after the other decimal, without any decimal point between them. Examples: .06.065 .83.875 .03.03125 To read complex decimals, it is necessary to use the word "and" just preceding the fractional part of the decimal. This, however, does not indicate the decimal point. Example: .06 is read "six and one-half hundredths." 3.06 is read "three and six and one-half hundredths.” = 45. 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. 9. 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. 9, 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 10, 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 turn. One of these divisions on e will, therefore, indicate of a turn, and the distance represented will be of 1000 in. The micrometer measurement, or the distance between c and a, is indicated by the length of scale which is exposed on d; that is, the distance between the zero point and the edge of the thimble. To read the micrometer is simply a matter of reading this length. The large numbered divisions, as explained, are equal to tenths of an inch, so we first set down the numbers of tenths of an inch, as shown by the last number exposed on the barrel d. The small divisions are equal to .025 in., so for each additional small division entirely exposed to the right of the last tenths mark, we add .025 to the number of tenths. The number of thousandths corresponding to the last small division which is only partly exposed is read from the scale on the thimble at the point where the two scales intersect. This added to the rest gives the complete measurement. Example: .7 .025 .018 Let us read the micrometer shown in Fig. 9. .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. Many 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 = 5 X .125.625; and 7 X .125 .875, etc. Also, if possible, = = |