name, then divide by such number as will reduce it to the next greater, always annexing ciphers to the dividend, as occasion may require : thus proceed till it be reduced to the decimal of the required integer. Or, reduce the given parts to a simple quantity, by reducing them to the lowest name mentioned ; annex ciphers thereto, and divide by suchi numbers as will reduce them to the name required. Or, reduce the given parls to a vulgar fraction, and that fraction to a decimal. Example 1.-Reduce 178. 104d. to the decimal of a pound sterling. 1,9=.5+10d. = 10.800 =.875+178.= 7,875 =.89375. the decimal required. EXAMPLE 2.-Reduce 2 feet 9 inches to the decimal of a yard. Vulgar fraction. it, and 33.0000 9:0000 = .9166 as required. To find the value of any given decimal. RULE.-Multiply the decimal given by the number of parts of the next inferior denomination, cutting off the decimals from the product ; then multiply the remainder by the next infe denomination; thus proceeding till you have brought the least known parts of the integer. EXAMPLE 1.—Required the value of .89375 of a pound sterling. .89375 20 17.87500 12 10.50000 2 1.00000 or, 178. 104d, Example 2.-Reduce .625 of a hundred weight to its proper terms. .625 X 4 = 2.500 x 28 = 14.000, or quarters and 14 lbs. ADDITION OF DECIMALS. Rule.-Arrange the numbers under each other, according to their several values ; find the sum, as in addition of whole numbers, and cut off for decimals as many figures to the right hand as there are decimals in any one of the given numbers. EXAMPLE.—What is the sum of 23.45, 7.849, 543.2, 8.6234, 253.004 ? 23.45 If any of the decimals be repetends, 7.849 continue them beyond the others, and 543.2 make them end together; then in adding, 8.6234 253.004 increase the sum of the first column by as many units as there are nines therein; as, 836.1264 .75 Here the first sum 18 contains two .6666 nines; therefore 2 added to 18 = 20. The .8888 rest of the work is the same as usual in .875 .4444 others; the repetend is 0, so the sum is finite. 3.6250 If some of the decimals be repetends, and others circulates, continue them both beyond those that are finite, and till their periods end together; then to the sum of the first column add as many as would arise to carry to it if they were continued farther; so will you have a circulate in the sum. Thus, 2.5 The repetend of .6, the circulate of 3.666666 69 and .372, continued till their periods 7.696969 14.372372 end together. It may easily be observed that there would be i to carry to the first 28.236008 column if it were carried any farther. Note. It is not always necessary to attend to the rule for repetends and circulates; three or four decimal figures, according to the rule, being sufficiently near the truth for common calculations. SUBTRACTION OF DECIMALS. Rule.-Place your numbers according to their value, subtract as in whole numbers, and cut off for decimals, as in addition. EXAMPLE.-Subtract 35.87043 from 132.005. 132.005 If both be single repetends, make them 35.87043 end together; and if there be occasion to borrow at the first figure, borrow 9 96.13457 only instead of 10; thus,-.8333 If both be circulates, or one a re.6666 petend and the other a circulate, con tinue both till their periods end toge.1666 ther; then if there should be occasion to borrow at the following figure, were they continued that figure farther, carry 1 to the first figure; and if the numbers be in different denominations, reduce them till they be alike. Subtract $39 from lý; thus, 1.666666 .834834 .831831 MULTIPLICATION OF DECIMALS. Rule. Place the factors under each other, and multiply them together, as in whole numbers; then point off as many figures from the product (counting from right to left) as there are decimal places in both factors; observing, if there be not enough, to annex as many ciphers to the left hand of the product as will supply the deficiency. EXAMPLE.-Multiply .4375 by .125. .4375 Here the product of .4375 by .125 is .125 546875; but as there are three places of decimals in the multiplier, and four in the 21875 multiplicand, a cipher must be added on 8750 4375 the left hand of the product to reduce it to its proper terms. .0546875 To multiply a repetend by a single figure, add 1 to the first product for every 9 therein, so will you have a repetend in the product; and if there be several figures in the multiplier, do so with each product, and continue them till they end together; then add them as so many repetends. If the multiplicand be a circulate, consider the increase that would arise to the first product if the multiplicand was continued farther : thus do with each product, make them end together, and add them by the rule for adding circulates. To Contract the operation, so as to retain only as many Decimals in the Product as may be thought necessary. RULE. Place the unit figure of the multiplier under that figure of the multiplicand whose place is the last to be retained in the product, and dispose of the rest so that they may all stand in contrary order to that in which they are usually placed. Then, in multiplying, reject all the figures to the right hand of the multiplying digit, and set down the product, so that the right hand figures may fall in a straight line under each other; observing to increase the first figure of every line with what would arise by carrying 1 from 5 to 14,-2 from 15 to 24,-3 from 25 to 34, &c., from the product of the two preceding figures, when you begin to multiply; and the sum will be the product required. EXAMPLE.-Multiply 27.14986 by 92.41035. Common way. Contracted way: 27.14986 27.11986 92.41035 53014.29 131574930 8144958 27149860 108599 44 5429972 24434874 24434874 542997 81 2508.9280 650510 2508.9280 DIVISION OF DECIMALS. Rule.--Prepare your decimals as directed for multiplication, divide as in whole numbers, cut off as many figures for decimals in the quotient as the number in the dividend exceeds the number in the divisor, namely, make the number of decimal figures in the divisor and quotient together equal to the number in the dividend. EXAMPLE.-Divide 173.5425 by 3.75. 3.75)173.5425( 46.278 Although you may take ad150.0 ditional ciphers at pleasure, 2354 care must be had in reckoning 2250 the number taken in dividing for decimals in the dividend; 1042 750 and if you put the decimal point in the quotient at any 2925 part of the operation, continu2625 ing the operation afterwards 3000 will not cause the point to be 3000 removed. |