in a contrary order, and the units place (7) is under (4) the 3d decimal from the left, the figures in the multiplier will stand directly under those in the multiplicand where the respective multiplications must begin. *5849 62.7 4094 35 4246 Here 6 is carried to 7 times 4, because 7 times 9 (the figure omitted) is 63;-1 is carried to 2 times 8, because twice 4 (the figure on the right of 8) exceeds half 10.—And 5 is carried tó 6 times 5, since 6 times 8 make almost 5 tens. As a further illustration of this method of contraction, take the following examples. Multiply 8467-73912 by 0725181, reserving only 4 decimals in the product. 8467-73912 481527 0 Here (0) the units place stands under the 4th decimal from the left. 59274174 1693548 423387 8468 6774 338 6140-6689 Product. Multiply 3842-63 by 79-6313, retaining the integers only. Here units stand under units, no decimals being required in the product. 3842-63 3456-97 268984 34584 2306 192 15 306082 Product. DIVISION OF DECIMALS. 65. DIVIDE as in whole numbers and point off as many decimals in the quotient as the number of decimals in the dividend. exceed those in the divisor. But if the number of figures in the quotient are not so many as the rule requires, prefix ciphers on the left to supply the defect. If the number of decimals in the divisor exceed those in the dividend, annex ciphers to the latter before you begin the division. - When the divisor is 1 with ciphers on the right hand, remove the decimal point in the dividend as far to the left as there are ciphers. But when the divisor is any other number with ciphers annexed, first divide by 10, 100, or 1000, &c. according to the number of ciphers; then divide the quotient by the remaining figure or figures. (60) N. B. Should there be a remainder after division, ciphers may be annexed to it, and the division continued as far as is necessary. Product 01728 Hence it appears that the number of decimals in the divisor and quotient must be equal to those in the dividend; and therefore the truth of the rule is manifest. 66. When a certain number of decimals only are wanted in the quotient, the division may be contracted in the following manner: Take the divisor one figure more than the number of figures required to be in the quotient. Make each remainder a new dividend, and for every such dividend leave out a figure on the right hand of the divisor, remembering to carry for the increase of the figures omitted as in the contraction of multiplication. (64) Let 94-78 be divided by 2.84671281 so as to have 4 decimals in the quotient. The number of figures in the quotient will be 6, viz. 2 integers and 4 decimals, therefore we must take 7 figures for the divisor. 2-84671281) 94-78000 (33.2945 quotient. 8540138 2S4671) 957862 854014 28467) 83848 2846) 26911 25620 281) 1294 1138 21) 156 142 14 The two right hand figures (81) of the given divisor are cut off, and 2 are carried for the product of S by 3. And instead of bringing down each divisor (as above) the figures may successively be pointed off. It is also evident when the number of figures in the divisor is less than the number required in the quotient, that ciphers must be added to the former. To reduce a Vulgar Fraction to an equivalent Decimal. 67. ADD ciphers to the numerator and divide by the denominator, then point off as many decimal places in the quotient for the answer as there were ciphers annexed. This is con. tinuing the division of whole numbers when there is a remainder, by which micans we get a decimal in the quotient instead of a vulgar fraction. For example, if 97 be divided by 31, the quotient is or 3, but if ciphers are added we shall have 3-03125 for the quotient. Thus, 32) 97.00000 (3-C3125 quotient, 96 100 96 40 32 08 64 160 The ciphers annexed only point out the number of decimal places, and therefore 97-00000 is the same as 97, consequently 97-00000 and 97 when divided by 32 must give equal quotients, and therefore the decimal 03125 is equivalent to. Which also will be evident by taking the decimal as 2 vulgar fraction, for reduced to its lowest terms is 3125 98 Divide 98 by 24000; or which is the same thing, reduce to a decimal? The decimals in the last examples are called circulating, recurring, or repeating decimals, because the same figure or figures are regularly repeated. 68. When an improper fraction is to be reduced, the answer will be a mixt number. Thus 925; and 12.666 &c. 766 |