Another way to know the place for the decimal point, is this: The first figure of the quotient must be made to occupy the same place, of integers or decimals, as doth that figure of the dividend which stands over the unit's figure of the first product. When the places of the quotient are not so many as the Rule requires, the defect is to be supplied by prefixing ciphers. When there happens to be a remainder after the division'; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the divi. dend, and the quotient carried on as far as required. EXAMPLES. 1. 2. 178) •48520998 (:00272589 -2639 ) 27600000 (1023114 1292 6100 460 8220 1049 3030 1599 3910 1758 12710 156 2154 When the divisor is an integer, with any number of ciphers annexed : cut off those ciphers, and remove the deci. mal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before *. * This is no more than dividing both divisor and dividend by the same number, either 10, or 100, or 1000, &c, according to the number of ciphers cut off, which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same number of places in both the divisor and dividend, either to the right or left, whether they have ciphers or not. EXAMPLES EXAMPLES. 1. Divide 45.5 by 2100. 21:00) .455 (0216, &c. 35 2. Divide 41020 by 32000. 3. Divide 953 by 21600, 4. Divide 61 by 79000. CONTRACTION II. Hence, if the divisor be 1 with ciphers, as 10, 100, or 1000, &c: then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left, as the divisor has ciphers; prefixing ciphers if need be. EXAMPLES. And 419 = 10= And .21 ; 1000 = CONTRACTION III. WHEN these are many figures in the divisor; or when only a certain number of decimals are necessary to be retained in the quotient; then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder bę a new dividend, and for every such dividend, leave out one figure more on the right-hand side of the divisos; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication. Note. When there are not so many figures in the divisor, as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient; after which begin the contraction. EXAMPLES. 1. Divide 2508o92806 by 92.41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures. Contracted. Contracted. Common. 02:4103,5)2508.928,06(27.149892-4103,5)2508.928,06(27.1498 660721 66072106 13849 13848610 4608 46075750 912 91116100 80 79167850 6 5539570 2. Divide 4109.2351 by 230•409, so that the quotient may contain only four decimals. Ans. 17.8345. 3. Divide 37•10438 by 5713.96, that the quotient may contain only five decimals. Ans. •00649. 4. Divide 913.08 by 2137.2, that the quotient may contain only three decimals. REDUCTION OF DECIMALS. CASE I. To reduce a Vulgar Fraction to its equivalent Decimal. Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required. EXAMPLES. 1. Reduce to a decimal. 26) 1.750000. .291666 &c. 2. Reduce , and, and to decimals. Ans. .25, and 5, and 75. 3. Reduce to a decimal, Ans. •625, 4. Reduce z's to a decimal. Ans. •12. 5. Reduce róz to a decimal. Ans..031359. 6. Reduce Ytr to a decimal. . Ans. •143155 &c. CASE 1 CASE II. To find the Value of a Decimal in terms of the Inferior Denisa minations. MULTIPLY the decimal by the number of parts in the next lower denomination ; and cut off as many places for a remainder to the right-hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations separated on the lefthand, will make up the answer. Note, This operation is the same as Reduction Descending in whole numbers. EXAMPLES. 1. Required to find the value of 775 pounds sterlinga •775 20 2. What is the value of .625 shil? Ans. 74d. 3. What is the value of •86351? Ans. 17s 3.24d. 4. What is the value of .0125 lb troy? Ans. 3 dwts. 5. What is the value of .4694 lb troy? Ans. 5 oz 12 dwts 15.744 gr. 6. What is the value of .625 cwt ? Ans. 2 qr 14 lb. 7. What is the value of .009943 miles ? Ans. 17 yd i ft 5.93848 inc. 8. What is the value of .6875 yd? Ans. 2 qr 3 nls. 9. What is the value of .3375 acr? Ans. I rd 14 poles. 10. What is the value of 2083 hhd of wine? Ans. 13.1229 gal. CASE CASE III. To reduce Integers or Decimals to Equivalent Decimals of Higher Denominations. Divide by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers. EXAMPLES. 1. Reduce 1 dwt to the decimal of a pound troy. 20 I dwt 0.05 oz 2. Reduce 9d to the decimal of a pound. Ans. .03751. 3. Reduce 7 drams to the decimal of a pound avoird. Ans, .027343751b. 4. Reduce •26d to the decimal of a l. Ans. .0010833 &c.l. 5. Reduce 2:15 lb to the decimal of a cwt. Ans. •019196 + cwt. 6. Reduce 24 yards to the decimal of a mile. Ans. .013636 &c. mile. 7. Reduce '056 pole to the decimal of an acre. Ans. .00035 ac. 8. Reduce 1'2 pint of wine to the decimal of a hhd. Ans. .00238 + hhd. 9. Reduce 14 minutes to the decimal of a day. Ans. .009722 &c. da. 10. Reduce .21 pint to the decimal of a peck. Ans. .013125 pec. 11. Reduce 28" 12" to the decimal of a minute, Note, When there are several numbers, to be reduced all to the. decimal of the highest : Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest. Opposite to each dividend, on the left-hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends. Begin at the uppermost, and perform all the divisions : only observing to set the quotient of each division, as decimal parts, |