NOTE. The number of figures of which a repetend will consist may be discovered by dividing 1 with ciphers annexed by the factors other than 2 or 5 of the denominator, until there is a remainder 1. Thus, if it be required to discover the number of figures in the repeating part of the decimal equivalent to, we divide 1 with ciphers annexed by 7, the only prime factor in the denominator other than 2 or 5, until there is a remainder of 1, which occurs after the sixth division, thereby indicating that the repeating part will consist of six figures. We have seen that these must be preceded by two places of finite decimals, so that the mixed repetend equal to must consist of eight places in all. 47 350 EXAMPLES. 3. To what kind of a decimal can be reduced? Ans. A pure repetend, of 2 places. 4. How many places of decimals, finite and repeating, will be required to express ? 39? Ans. 5 places; 3 finite and 2 repeating. 5. To what kind of a decimal can 103 be reduced? 6. Reduce 1313 to a mixed repetend. 166 Ans. 13.37. 7. Change to a mixed repetend. Ans. .008497133. 8. Of how many figures will the repetend consist that corresponds to? Ans. 28 figures. TRANSFORMATION OF REPETENDS. 298. Any finite decimal may be considered as a mixed repetend by making ciphers continually recur; thus, .42 = .420 .4200 = .42000, &c. 299. Any circulating decimal may be transformed into another having the same number of repeating figures; thus, .127 .1272.12727, &c. = 300. Any circulating decimal having as repetend any number of figures may be transformed to another having twice or thrice that number of figures, or any multiple thereof; thus, .5925, having a repetend of three figures, may be transformed .5925925 to one having 6, 9, 12, &c. places; therefore .5925 = 59259259255925925925925925, &c. = 301. The value of a decimal is not changed by any of the above transformations, as may be seen by reducing the given epetends to their equivalent common fractions (Art. 296) and comparing them together. Hence, they can be used in making lissimilar repetends similar and conterminous. 302. To make any number of dissimilar repetends similar and conterminous. Ex. 1. Make similar and conterminous 9.167, 14.6, 3.165, 12.432, 8.181, and 1.307. We make the finite mixed decimal, 14.6, a mixed repetend by annexing recurring ciphers, and make it and all the given repetends similar, by extending the figures to the right, so that the circulating part of each may begin at the same distance from the decimal point as does that repetend which is preceded by the most finite decimal places. Then, to make conterminous the repetends that have thus been rendered similar, as some of them consist of 1, some of 2, and the others of 3 places, we extend the repeating figures of each repetend till those of each occupy as many places as there are units in the least common multiple of 1, 2, and 3, which is 6. Hence, to make dissimilar repetends similar and conterminous, Transform the given repetends so that the circulating parts shall commence at the same distance from the decimal point, and shall consist of as many circulating places as there are units in the least common multiple of the number of repeating figures found in the given decimals. EXAMPLES. 2. Make 3.671, 1.0071, 8.52, and 7.616325 similar and con terminous. 3. Make 1.52, 8.7156, 3.567, and 1.378 similar and conter minous. 4. Make .0007, .141414, and 887.i similar and conterminous. terminous. ADDITION OF CIRCULATING DECIMALS. 303. Ex. 1. Add 2.765, 7.16674, 3.671, .7, and .1728 the right. In that case we should have had to carry 3 after finding the amount of the first left-hand column of the repetends continued. We therefore increase the sum as first found, and thus have the true amount as in the operation, 14.55436. RULE. - Make the given repetends, when dissimilar, similar and conterminous. Add as in addition of finite decimals, observing to increase the repetend of the amount by the number, if any, to be carried from the left-hand column of the repetends. EXAMPLES. 2. Add 3.5, 7.65i, 1.765, 6.173, 51.7, 3.7, 27.631, and 1.003 together. Ans. 103.2591227. 3. Reduce, and to decimals, and find their sum. 4. Find the sum of 27.56, 5.632, 6.7, 16.356, .71, and 6.1234. Ans. 63.1690670868888. 5. Add together.165002, 31.64, 1.06, .34634, and 13. 6. Add together .87, .8, and .876. Ans. 2.644553. 7. Required the value of .3 .45 + .45 + .351 +.6468 +.6468.6468, and .6468. Ans. 4.1766345618. 8. Find the value of 1.25 +3.4.637 +7.885 +7.875 +7.875 + 11.i. Ans. 40.079360724. 9. Add together 131.613, 15.001, 67.134, and 1000.63. 10. Find the value of 5.16345 +8.6381 +3.75. Ans. 17.55919120847374090302. SUBTRACTION OF CIRCULATING DECIMALS. 304. Ex. 1. From 87.1645 take 19.479167. Dissimilar. OPERATION. = 19.47 9167 = Similar and Conterminous. 87.1 6 4 5 87.1 645 45 Ans. 67.685377 Ans. 67.685377. Having made the repetends similar and conterminous, we subtract as in whole numbers, regarding, however, the right-hand figure of the subtrahend as increased by 1, since 1 would have been carried to it in subtracting, if the repetends had been continued farther to the right, as is evident from the circulating part of the subtrahend being greater than that of the minuend. RULE. Make the repetends, when dissimilar, similar and conterminous. Subtract as in subtraction of finite decimals; observing to regard the repetend of the subtrahend as increased by 1, when it exceeds that of the minuend. 2. From 7.1 take 5.02. EXAMPLES. 3. From 315.87 take 78.0378. Ans. 237.838072095497. 4. Subtract from . Ans. 2.08. Ans. .079365. 5. From 16.1347 take 11.0884. 6. From 18.1678 take 3.27. 7. From 3.123 take 0.71. 8. From take fi 9. From take &. 10. From take 167. 11. From 5.12345 take 2.3523456. Ans. 2.7711055821666927777988888599994. MULTIPLICATION OF CIRCULATING DECIMALS. 305. Ex. 1. Multiply .36 by 25. reduced to its equivalent decimal, gives .0925, the answer required. RULE. Change the given numbers to their equivalent common fractions. Multiply them together, and reduce the product to its equivalent decimal. EXAMPLES. 2. Multiply 87.32586 by 4.37. 3. Multiply 582.347 by .03. Ans. 381.6140338. Ans. 13.5169533. 5. What is the value of .285714 of a guinea? Ans. 8s. 6. What is the value of .461607142857 of a ton? Ans. 9cwt. Oqr. 23+lb. 7. What is the value of .284931506 of a year? mon fractions, and, dividing, obtain 27, which, reduced to its equivalent decimal, gives 3.506493, the answer required. RULE. ·Change the given numbers to their equivalent common fractions. Divide, and reduce the quotient to its equivalent decimal. 2. Divide 345.8 by .6. 3. Divide 234.6 by .7. EXAMPLES. 4. Divide 13.5169533 by 3.145. 6. Divide .428571 by .625. 7. Find the value of 2.370 ÷ 4.923076. 9. Find the value of 316.31015 ÷ .3. 10. Find the value of 100006 ÷ .6. Ans. 518.83. Ans. 4.297. Ans. 87.32586. Ans. .481. Ans. .39. 11. Divide .36 by .25. Ans. 1.4229249011857707509881. |