because there are as many in the multiplicand and multiplier. The same reasoning applies to all multiplications of decimals. 0,036 is the answer. We first multiply 12 by 3, which gives 36 for the product. The rule directs to point off three figures for decimals, but there are only two in the product. By applying the reasoning of the preceding case, this embarrassment is easily removed, for it requires the interposition of a cipher between 36 and the separatrix. Because if 0,12 is multiplied by 3, the product is evidently 0,36; but since we multiply by 0,3, that is, by a number ten times smaller than 3, the product must be ten times smaller than 0,36, or it will be thousands; and this value is expressed by writing it (28) 0,036. EXAMPLES FOR PRACTICE IN THE MULTIPLICATION OF DECIMALS. 1. Multiply 223,86 by 2,500. 2. Multiply 35,640 by 26,18. 3. Multiply 8,4960 by 2,618. 4. Multiply 0,5236 by 0,2808. 5. Multiply 0,11785 by 0,27. Ans. 559,65000. Ans. 933,05520. Ans. 0,0318195. 55. As the ordinary use of decimals is to facilitate calculation, by substituting instead of an exact computation a sufficient and ready approximation; a rule is here given for abridging the operation when the product is required only to a certain degree of exactness. For example, if 45,625957 is to be multiplied by 28,635, and the decimal part of the product is required only to thousandths of an unit; the numbers are written as below; that is, after having reversed the order of the figures in one of the factors, we write it under the other, and make its unit figure correspond to the decimal figure in the multiplicand, which is removed two places towards the right from the figure, which has the value of that to which we intend to limit the product. Multiplication is then performed, by neglecting, in the multiplicand, all the figures to the right of that by which we multiply; and the products found by multiplying by the several figures of the multiplier, are written vertically under each other. These products are then added together and two figures suppressed on the right; but the last of the remaining figures is augmented by unity, when the two suppressed figures exceed 50. The separatrix is then put in the place required by the kind of decimals proposed. EXAMPLE. We wish to multiply 45,625957 by 28,635 but do not require the decimal part of the product nearer than to thousandths of an unit. The numbers are written in this manner. 45,625957 53682 91251914 36500760 2737554 136875 22810 130649913 Product 1306,499 If multiplication had been made in the ordinary manner, the product would be 1306,499278695, which corresponds with the preceding up to the third decimal, as was required. When there are not decimal places enough in the multiplicand to allow the unit figure in the multiplier to correspond to the figure which the rule requires, ciphers may be put to supply the deficiency. USES OF MULTIPLICATION. 56. We propose to show some uses to which multipli cation may be applied, by furnishing such examples as will direct the learner in solving questions of a similar nature. In general, multiplication is useful for finding the total value of many units, when the value of each separate unit is known. For example, how much will 5842 acres of land come to, at 54 dollars an acre? 54 dollars must be multiplied by 5842, or (44) 5842 dollars by 54; and $15468 dollars will be the whole price demanded. The following are the Tables of Units referred to in the preceding examples. The denominations in these Tables will be subsequently recurred to. Example 2. How much will 5954 cubic feet of water weigh, 72 pounds being the weight of one foot? 72 pounds must be multiplied by 5954, or 5954 by 72; and 428688 pounds will be the weight of 5954 cubic feet. IN CUBIC OR SOLID MEASURE, 1728 solid inches make 128 solid feet 40 solid feet of round timber or 50 solid feet of hewn timber 1 solid foot. 1 cord of wood. 1 ton or load. IN AVOIRDUPOIS WEIGHT, 16 drams (dr.) make 1 ounce, marked oz. 1 pound, 1 quarter, lb. qr. 1 hundred weight, cwt. 1 ton, T. By this weight are weighed all heavy goods, except gold, silver, corn, bread, and liquors, which are weighed by Troy weight. 57. Multiplication may be used for reducing units of one denomination into units of a lower denomination.For example; dollars may be reduced to dimes, cents, &c.; and tons to pounds, ounces, &c. by multiplying the units of the higher denomination by the number of the next lower which expresses an unit in this higher denomination, and adding to the product the units, if there be any, of the same name. Thus, if the number of pence in 8 pounds, 17 shillings and 7 pence is required; we seek in the following Table for the number which expresses the shillings in a pound, and find that it is 20. And the operation is commenced by multiplying £8 by 20 (52,) which gives 160s. for the product, to which add the 17s. in the sum, and the whole number of shillings becomes 177; which is multiplied by 12, because 12 pence make 1 shilling, and the product is 2124 pence, to which add the 7 pence in the sum, and 2131 is the whole number of pence contained in £8 17s. 7d. MEASURE OF TIME. 60 seconds (s. or ") make 1 minute, marked m. or If the number of minutes in a common year of 365 days, 5 hours, 48 minutes, is required; these numbers are written in this manner, D. h. m. $65 5 48 24 1465 730 8765 60 525948 Because 24 hours make 1 day, 24h. are multiplied by 365 and the product is 8760h., to which the 5h. are added, and 8765, the product, is multiplied by 60 (52,) because the hour contains 60 minutes, and 525900 m. is the product, to which add 48m. and 525948 will be the number of minutes contained in the common year. TROY WEIGHT. 24 grains (grs.) make 1 penny weight, marked dwt. 20 dwt. 12 oz. 1 ounce, By this weight are sold gold, silver, corn, bread, and liquors. |