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 ,47 3309 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 15 must consist of eight places in all. EXAMPLES 3. To what kind of a decimal can it be reduced ? Ans. A pure repetend, of 2 places. X 4. How many places of decimals, finite and repeating, will be required to express 1330? Ans. 5 places ; 3 finite and 2 repeating. 5. To what kind of a decimal can 3:} be reduced ? 6. Reduce 131} to a mixed repetend. Ans. 13.37. 7. Change 136lto a mixed repetend. Ans. .008497133. 8. Of how many figures will the repetend consist that corresponds to ? Ans. 28 figures. TRANSFORMATION OF REPETENDS. S 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 nunber 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 to one having 6, 9, 12, &c. places; therefore .5925 = .5925925 5925925925 = 5925925925925925, &c. 301. The value of a decimal is not changed by any of the above transformations, as may be seen by reducing the given repetends to their equivalent common fractions (Art. 296) and comparing them together. Hence, they can be used in making dissimilar repetends similar and conterminous. RULE. — Change the given numbers to their equivalent common fractions. Multiply them together, and reduce the product to its equivalent decimal. 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. OPERATION Dissimilar. Similar. Similar and Conterminous. 9.1 6 7 6 7 6 7 6 1 4.6 1 4.6 0 0 1 4.6 0 0 0 0 0 0 0 3.1 6 5 5 5 5 5 5 1 2.4 3 2 Ans. 1 2.4 3 2 4 3 1 2.4 3 2 4 3 2 4 3 8.1 si 8.1 8 is 8.1 8 81815 1.3 0 7 1.3 0 1 3 0 1.3 0 7 3 0 7 3 0 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 3.67i, 1.007i, 8.52, and 7.61632) similar and conterminous. 3. Make 1.52, 8.7156, 3.567, and 1.378 similar and conterminous. 4. Make .0007, .141414, and 887.i similar and conterminous. 5. Make .3123, 3.27, and 5.02 similar and conterminous. 6. Make 17.0884, 1563.0929, and 15.12345 similar and conterminous. ADDITION OF CIRCULATING DECIMALS. add as 303. Ex. 1. Add 2.765, 7.16674, 3.671, .7, and .i728 together. Ans. 14.55436. OPERATION. Having made the given Dissimilar. Similar and Conterminous. repetends similar and con2.7 6 5 2.7 6 5 65 terminous (Art. 302), we 7.1 6 67 4 7.1 6674 in addition of 3.6 7 i whole numbers, and obtain 3.6 7 1 3 6 14.55433. The right-hand . .71771 figure of this result we in.i 7 2 8 .11 28i crease by such a number as would have been carAns. 1 4.5 5 4 3 6 ried, if the repetends had been continued farther to 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, 27.63i, and 1.003 together. Ans. 103.2591227. 3. Reduce }, }, and f to decimals, and find their sum. 4. Find the sum of 27.56, 5.632, 6.7, 16.356, 7i, and 6.1234. Ans. 63.1690670868888. 5. Add together.165002, 31.64, 1.06,.34634, and 13. 6. Add together .87, .s, 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, 15.000, 67.134, and 1000.63. 10. Find the value of 5.16345 + 8.6381 +3.75. Ans. 17.55919120847374090302. SUBTRACTION OF CIRCULATING DECIMALS. Dissimilar. OPERATION, Similar and Conterminous. 304.° Ex. 1. From 87.1645 take 19.479167. Ans. 67.685377. Having made the repe tends similar and conter8 7.1 6 4 5 87.1 6 4 5 4 5 minous, we subtract as in 1 9.4 7 9 167 1 9.4 7 9161 whole numbers, regarding, however, the right-hand Ans. 6 7.6 8 5 3 7 7 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. EXAMPLES. 2. From 7.i take 5.02. Ans. 2.08. 3. From 315.87 take 78.0378. Ans. 237.838072095497. 4. Subtract from . Ans. .079365. 5. From 16.1347 take 11.0884. Ans. 5.0462. 6. From 18.1678 take 3.27. Ans. 14.8951. 7. From 3.123 take 0.7i. Ans. 2.405951. 8. From , take si: Ans. .246753. 9. From take 4. Ans. .i58730. 10. From i, take 18 Ans. .i76470588235294i. 11. From 5.12345 take 2.3523456. Ans. 2.7711055821666927777988888599994. OPERATION. MULTIPLICATION OF CIRCULATING DECIMALS. 305.° Ex. 1. Multiply .36 by 25. Ans. 0929. We change the 36 4 .36 23 25 to given numbers their equivalent com- mon fractions, and, A X 3 = .0929 Ans. multiplying, obtain reduced to its equivalent decimal, gives .0925, the answer required. ܪ 92 46 495 EXAMPLES. 2. Multiply 87.32586 by 4.37. Ans. 381.6140338. 3. Multiply 582.347 by .03. 4. Multiply 3.145 by 4.297. Ans. 13.5i 69533. 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? Ans. 104d. DIVISION OF CIRCULATING DECIMALS. OPERATION. ;.15 306.° Ex. 1. Divide .54 by .15. Ans. 3.506493. We change 54 6 15 14 .54 7 the given num bers to 99 their 11 10 45 90 equivalent com= 75 3.506493 Ans. mon 278 fractions, and, dividing, obtain 470, 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. EXAMPLES. 2. Divide 345.8 by ... Ans. 518.83. 3. Divide 234.6 by .7. 4. Divide 13.5i69533 by 3.145. Ans. 4.297. 5. Divide 381.6140338 by 4.37. Ans. 87.32586. 6. Divide .428571 by .625. 7. Find the value of 2.370 + 4.923076. Ans. .48i. 8. Find the value of .09 • .230769. Ans..39. 9. Find the value of 316.31015 : .3. 10. Find the value of 100006 : ... 11. Divide .36 by .25. Ans. 1.422924901185770750988i. |