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eramples. Multiply 741,864 by 100 Multiply 6,417101 by 1000. - 74186,4 6417,101 . Multiply 635,4 by 100 Multiply 5846,5 by 1000 63540 - 5846500 1. 1075,426 × 10 2. . 2641,07 x 100 3. 17,5 × 100 4. 617,5 × 1000
5. 800,95 x 10000 6. 36,742073 x 100000
When one decimal is to be multiplied by another, and that it is not necessary to retain all the decimal places in the product; write the units place of the multiplier, under that figure of the multiplicand, whose place it would be desirable to retain, and set down the rest of the figures in an inverted order.
Reject all the figures that are to the right of each multiplying figure, and place the products, so that their right-hand figures may be in a perpendicular line under each other, observing to increase the first figure of every product, with what would arise by the multiplication of the figures rejected, by carrying l from 5 to 15; 2 from 15 to 25; 3 from 25 to 35 &c. the sum will be the product required, sufficiently near for most purposes. Cramples: Multiply 741,042 by 46,372 Multiply 142,0059 by 3,7584 retaining but 3 decl. places. retaining but 4 decl. places.
741,042 741,042 142,0059 . 142,0059 273,64. 46,372 48573° 3,7584 296.41680 1482084 4260.177 5680236 : 4446252 - 5187294 99.404 lo 1-1360472 222313 - 2223126 71003 . 7100295 51872 4446252. 11360. 9940413 1482 296.4168 568 . . .4260177
Multiply 64;1976 by 641967 retn. 3, 4 and 5 decl places. . Multiply 1,33333 by ,5236 retn. 3, 4 and 5 decl places. Multiply 39,3332 by ,07958 retu. 3, 4 and 5 decl places. Multiply 698,23728 by ,7854 retn. 3, and 4 decl places. . Multiply 1378,5754 by 1,7321 retaining 4 decimal places. . Multiply 49912,34 by 3,1416 retaining 4 decimal places. ... Multiply 796,042 by 12,5644 retaining 4 decimal places. . Multiply 406,253 s by 62,8327 retn 3, 4 and 5 deel places.
, DIVIVISION OF DECIMALS.
Proceed as in whole numbers, and from the right-hand point off as many places for decimals in the quotient as those in the dividend exceed those in the divisor.
, If the decimal places in the quotient are not as many as the rule requires, supply the deficiency by prefixing ciphers.
If there should be a remainder, or the decimal places in the divisor exceed those in the dividend, ciphers may be affixed to the dividend, and the quotient carried on as far as is necessary.
When the decimal places in the dividend and divisor are equal, the quotient will be a whole number, provided the divisor is a measure of the dividend; if there is a remainder, and that it is necessary to continue the division, annex to the whole number already found, one decimal place for every cipher annexed to the dividend. -
In general, four or five places at most, will be as far as may be necessary to continue the division for any practical purpose, if it should not sooner terminate, as the quotient will thereby be within rows part of the truth. . .
When the divisor is 10, 100, 1000, &c. division is instantly performed, by removing the decimal point as many places
to the left, as the divisor hath ciphers, prefixing ciphers is necessary. -
Cramples. Let 36417,341 be divided by Let 7,384.107 be divided by 100 = 364,17341 - 1000 = 007384.107 I. 84968,43 - 10 v 2. 2758,496 -i- 100 3. 759,846 - 100 4. 876,498 -- 1000
5. 83,14786 -- 10000 6. 348,758 -- 10000
To contract the work, so as to retain but as many decimal places in the quotient, as may be thought necessary —
Take as many of the left-hand figures of the divisor as will be equal to the number of figures both integers and decimals, which are to be in the quotient, and find how many times they are contained in the first figures of the dividend as usual.
Let each remainder be a new dividend, and for every such dividend, leave out one figure more on the right-hand of the divisor, remembering to carry for the increase, as in the second contraction in multiplication.
When there are not as many figures in the divisor as are required, begin the operation with all the figures, and continue as usual, till the number of figures in the divisor are equal
to those remaining to be found in the quotient, after which, use the contraction.
Divide 3758,49645 by 58,3194, Divide 79,56876 by 7,49, and retain but 3 places of deci- and retain but 4 places for
mals in the quotient. decimals in the quotient. 58,3194)3758,496.45(64,446 7,43)7956876(10,6233. 25933 4688 2605 - 174 272 24 39 - 2 - 4 -
1, 254894,286 + 924}0,35, retaining 4 places of decimals. 2. 4109,2351 + 240,409, retaining 3 places of decimals. 3. 31710,438 -- 5713,96, retaining 5 places of decimals. 4. 9130,8 -- 2173,24, retaining 3 places of decimals. 5. 6710,984.75 -- 69,4875, retaining 4 places of decimals. 6. 54,98675 -- 375,849, retaining 3 places of decimals. 7. 121698,25 + 3141,59, retaining 5 places of decimals. 8, 671,49384 + 498,5395, retaining 4 places of decimals.
RULE OF PROPORTION:
Proportion is the equality of Ratios, and the terms of equal ratios are called Proportionals. Ratio, when applied to the subject of arithmetic, is the mutual relation of two numbers of the same kind to one another, . with respect to quantity; or is the comparative magnitude of two numbers of the same kind. Numbers are said to have ratio to each other when they are such, that the less can be multiplied, so as to exceed the greater. The two numbers compared are called the terms of the ratio, the first the antecedent, and the latter the consequent. Ratios are equal when similar parts of their antecedents are respectively contained the same number of times in their consequents, thus, the ratio of 6 to 8 is equal to the ratio of 9 to 12; because 2 the third part of 6, and 3 the third-part of 9, are contained, the one in 8, and the other in 12, each 4 times. And if each of these be expressed as a fraction, making the antecedents the numerators, and consequents the denominators, the expressions thus formed will be equal, that is 4-3 and +”,-3; in either cases, 6 is said to have to 8, the same ratio which 9 has to 12, or it is simply said, that 6 is to 9 as 8 is to 12. Ratio is expressed by the aid of 2 dots interposed between the terms of the ratio, and the identity or equality of two ratios by 4 dots interposed between the ratios, thus 6:8::9 : 12 denotes, that the ratio of 6 to 8 is the same or equal to the ratio of 9 to 12. Proportion consists of three terms at least. Three numbers are said to be proportional, when the ratio of the first to the second, is the same as the ratio of the second to the fourth ; as 2, 4, 8, where 4-#=2, both the same ratio, for 4 is as much greater in comparison of 2, as 8 is greater than 4. - - Four numbers are said to be proportional, when the ratio of the first to the second is the same as the ratio of the third to the fourth, as of 2:4:: 5: 10, where 3- $:=2, both the same ratio. Four numbers are also said to be proportional, when by comparing them together by two and two, they either give equal products or equal quotients; suppose the former numbers 2:4: 5 10; in comparing them together by multiplication, we have
therefore, 2 : 4:: 5 : 10 are called proportional numbers. , Now let the same numbers be compared by division, and we have
By this comparison the numbers are also said to be proportional. - In this kind of comparing four numbers, there is no need to try for more equal products or quotients than one set of each sort ; for either case will determine the proportionality independent of the other. But it must be observed, that in any 4 proportional numbers, there will be but one set of equal products, and 2 sets of equal quotients, the smaller numbers being divisors. When two numbers of the same kind, of different magnitudes are compared, it is instantly seen, that one of them is greater than the other; thus, if we compare 4 and 1, the former is evidently 4 times greater than the latter, and if each of these be multiplied by any common multiplier, as 2, the products 8, and 2 will bear the same comparative magnitudes to each other, as the original numbers, for in proportion as 4 exceeds 1, so will 8 exceed 2, and the numbers being placed, as under, will form what is called an analogy or proportion. - * , - 4 : 1 ::8 : 2. * The like will hold good, with any numbers or multipliers, and as it is evident, that this position is correct, with respect to the products, arising from a common multiplier; it is equally the case with respect to the quotients, arising from the division of two numbers by a common divisor; thus if we make choice of 4 and 10, and multiply and divide each of them by 2, by the multiplication there will result 8 and 20, and by the division 2 and 5, by which we may form two separate analogies, from the first - 4 : 10 : :8 : 20
2 : 5::4: 10
and from the second