CASE II. To reduce mixed circulating decimals to common fractions. 356. 11. Reduce .16 to a common fraction. Analysis.-Separating the mixed decimal into its terminate and periodical part, we have .16 .1+.06. (Art. 320.) Now .1=; (Art. 312;) and .06; for, the pure period .6=, (Art. 351,) and since the mixed period .06, begins in hundredths' place, its value is evidently only as much; but §÷10=%. (Art. 227.) Therefore .16+%. Now and, reduced to a common denominator and added together, make 15, or . Ans. OBS. In mixed circulating decimals, if the period begins in hundredths' place it is evident from the preceding analysis that the value of the periodical part is only as much as it would be, if the period were pure or begun in tenths' place; when the period begins in thousandths' place, its value is only as much, &c. Thus .6=9; .06=5÷10= £: .006=§÷100=,, &c. 6 90 part 357. Hence, the denominator of the periodical part of a mixed circulating decimal, is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. 12. Reduce .8567923 to a common fraction. 9999900. Solution. Reasoning as before .8567923-5+973336. Reducing these two fractions to the least common denominator, (Art. 261.) 1999)9 8499915 whose denominator is the same as that of the other. Contraction. 8500000 85 8499915 1st Nu. 67923 2d Nu. 8567838 9999900 Ans. 9999900 Now 8499915. 67923 -8567838 Ans. 9999900 9999900 9999900. To multiply by 99999, annex as many ciphers to the multiplicand as there are 9s in the multiplier, &c. (Art. 105.) This gives the numerator of the first fraction or terminate part, to which add the numerator of the second or periodical part, and the sum will be the numerator of the The denominator is the same as answer. that of the second or periodical part. Second Method. 8567923 the given circulating decimal. 85 the terminate part which is subtracted 8567838 the numerator of the answer. 8567838 Ans. 9999900 Note.-1. The reason of this operation may be shown thus: 8567923== 8500000+-67923 Now 8500000-85+67923 is equal to 8567923--85. 2. It is evident that the required denominator is the same as that of the periodical part; (Art. 357;) for, the denominator of the periodical part is the feast common multiple of the two denominators. Hence, 358. To reduce a mixed circulating decimal to a common fraction. Change both the terminate and periodical part to common fractions separately, and their sum will be the answer required. Or, from the given mixed periodical, subtract the terminate part, and the remainder will be the numerator required. The denominator is always as many 9s as there are figures in the period with as many ciphers annexed as there are decimals in the terminate part. PROOF.-Change the common fraction back to a decimal, and if the result is the same as the given circulating decimal, the work is right. 13. Reduce .138 to a common fraction. Ans. 125, or 5. 14. Reduce .53 to a common fraction. 15. Reduce .5925 to a common fraction. 16. Reduce .583 to a common fraction. 19. Reduce .5925 to a common fraction. 20. Reduce .008497133 to a common fraction. CASE II.-Dissimilar periodicals reduced to similar and conterminous ones. 359. In changing dissimilar periods, or repetends, to similar and conterminous ones, the following particulars require attention. 1. Any terminate decimal may be considered as interminate by annexing ciphers continually to the numerator. Thus .46= .460000, &c.=,460' 2. Any pure periodical may be considered as mixed, by taking the given period for the terminate part, and making the given period the interminate part. Thus .46-.46+.0046, &c. 3. A single period may be regarded as a compound periodical. Thus .3 may become .33, or .333; so .63 may be made .6333, or .63333, &c.. 4. A single period may also be made to begin at a lower order, regarding its higher orders as terminate decimals. Thus .3 may be made .33, or .3333, &c. 5. Compound periods may also be made to begin at a lower order. Thus .36 may be changed to .363, or .36363, &c.; or by extending the number of places .479 may be made .47979, or .4797979, &c.; or making both changes at once .532 may be changed to .5325325, &c. Hence, 360. To make any number of dissimilar periodical decimals similar. Move the points, so that each period shall begin at the same order a" the period which has the most figures in its terminate part. 21. Change 6.814, 3.26, and .083 to similar and conterminous periods. Operation. 6.814 6.81481481 3.26=3.26262626 .083=0.08333333 Having made the given periods similar, the next step is to make them conterminous. Now as one of the given periods contains 3 figures, another 2, and the other 1, it is evident the new periodical must contain a number of figures which is some multiple of the number of figures in the different periods; viz: 3, 2, and 1. But the least common multiple of 3, 2, and 1 is 6; therefore the new periods must at least contain 6 figures. Hence, 361. To make any number of dissimilar periodical decimals, similar and conterminous. Firsi make the periods similar; (Art. 360;) then extend the figures of each to as many places, as there are units in the least commom multiple of the NUMBER of periodical figures contained in each of the given decimals. (Art. 176.) 22. Change 46.162, 5.26, 63.423, .488, and 12.5, to similar and conterminous periodicals. Operation. 46.162-46.16216216 5.26262626 63.423 63.42342342 0.48666666 5.26 = .486 125 =12.50000000 The numbers of periodical figures in the given decimals are 3, 2, 3, and 1; and the least common multiple of them is 6. Therefore the new periods must each have 6 figures. 23. Make .27, .3, and .045 similar and conterminous. ADDITION OF CIRCULATING DECIMALS. Ex. 1. What is the sum of 17.23+41.2476+8.61+1.5+ 35.423? Operation. Dissimilar. Sim. & Conterminous. 17.23 =17.2323232 41.2476=41.2476476 8.61 = 1.5 = 8.6161616 1.5000000 35.423 35.4232323 Ans. 104.0193648 First make the given decimals similar and conterminous. (Art. 361.) Then add the periodical parts as in simple addition, and since there are 6 figures in the period, divide their sum by 999999; for this would be its denominator, if the sum of the periodicals were expressed by a common fraction. (Art. 355.) Setting down the remainder for the repeating decimals, carry the quotient 1 to the next column, and proceed as in addition of whole numbers. Hence, 362. We derive the following general RULE FOR ADDING CIRCULATING DECIMALS. First make the periods similar and conterminous, and find their um as in Simple Addition. Divide this sum by as mony 9s as there are figures in the period, set the remainder under the figures added for the period of the sum, carry the quotient to the me column, and proceed with the rest as in Simple Addition. OBS. If the remainder has not so many figures as the period, ciphers must be prefixed to make up the deficiency. 2. What is the sum of 24.132+2.23+85.24+67.6? 3. What is the sum of 328.126+81.23+5.624+61.6? 4. What is the sum of 31.62+7.824+8.392+.027 ? 5. What is the sum of 462.34+60.82+71.164+.35? 6. What is the sum of 60.25+.34+6.435+.45+45.24 ? 7. What is the sum of 9.814+1.5+87.26+0.83+124.09 ? 8. What is the sum of 3.6+78.3476-+735.3+.375+.27+ 137.4? 9. What is the sum of 5391.357+72.38+187.21+4.2965+ 217.8496+42.176+.523+58.30048 ? 10. What is the sum of .162+134.09+2.93+97.26+3.769230 +99.083+1.5+.814? SUBTRACTION OF CIRCULATING DECIMALS. Ex. 1. From 52.86 take 8.37235. Operation. 52.86-52.86868 8.37235 8.37235 44.49632 We first make the given decimals si ilar and conterminous, then subtract as n whole numbers. But since the period in the lower line is larger than that above it, we must borrow 1 from the next higher order. This will make the right hand figure of the remainder one less than if it was a terminate decimal. Hence, 363. We derive the following general RULE FOR SUBTRACTING CIRCULATING DECIMALS. Make the periods similar and conterminous, and subtrææt as in whole numbers. If the period in the lower line is larger than that above it, diminish the right hand figure of the remainder by 1. OBS. The reason for diminishing the right hand figure of the remainder by 1, if he period in the lower line is larger than that above it, may be explained thus: When the period in the lower line is larger than that above it, we must evidently borrow 1 from the next higher order. Now if the given decimals were extended to a second period, in this period the lower number would also be |