239. COMPOUND PROPORTION IN DECIMALS. RULE. Prepare the numbers (if they require it) as in the preceding rule, and work as in Compound Proportion in whole numbers". EXAMPLES. 1. If 7 men earn 91. 10s. 6d. in 10 days, what sum will 28 men earn in 314 days. 2. If 30l. in 20 months gain 10l. 5s. what sum will 201. gain in 10 months? 3. If 3yd. 2qr. In. of cloth that is yard wide cost 9s. 6d., what cost 4yd. 3qr. 2n. of yard wide, and of equal goodness? Ans. 19s. 6d. 4. Paid 31. 7s. 4d. for the carriage of 5cwt. 3qrs. 150 miles; what sum will pay for the carriage of 7cwt. 2qr. 25lb. 64 miles at the same rate? Ans. 11. 18s. 7d..0525. h This rule depends on the same principles with Compound Proportion in whole numbers. CIRCULATING DECIMALS. 240. Circulating, repeating, or recurring decimals are those in which one or more of the figures continually recur, and may be carried on indefinitely; the figures that recur are called repetends. 241. A pure repetend is a decimal in which all the figures recur; as .222 &c. .012012 &c. .153153 &c. 242. A mixed repetend is a decimal in which some of the figures do, and some do not, recur; as .5333 &c. .341212 &c. .419375375 &c. 243. A single repetend is that in which only one figure repeats, as .333 &c and is denoted by a point placed over the circulating figure, as 3. 244. A compound repetend is that in which the same figures repeat alternately, as .1212 &c. .345345 &c. and is expressed by a point over the first and last repeating figure, as 12....345 &c. 245. Similar repetends are those which begin at equal distances from the decimal mark; thus .2357, .471, and .493857, are similar. 246. Dissimilar repetends are those which do not begin at equal distances from the decimal mark; thus .23123 and .4531 are dissimilar. 247. Conterminous repetends are such as end at equal dis tances from the decimal mark; thus 232323 and 315315 are conterminous, as are .34517 and .82413. 248. Similar and conterminous repetends are such as begin at the same distance from the decimal mark, and end at the same distance; thus .785343434 and .000789789 are similar and conterminous, as are .12345 and .54321. REDUCTION OF CIRCULATING DECIMALS. 249. To reduce a pure repetend to its equivalent vulgar fraction. RULE I. Under the given repetend as a numerator write as many nines as the repetend has figures for a denominator. II. Reduce this fraction to its lowest terms, which will be the fraction required. EXAMPLES. 1. Required the values of .3 and .36 in vulgar fractions? 2. Reduce .234 and 341231 to equal vulgar fractions. 2 5 3. Reduce .6 and .45 to vulgar fractions. Ans. and 3 11 5 5. Reduce .135 and 769230 to fractions. Ans. and 370 37 481 If unity, with ciphers subjoined, be divided by 9, in infinitum, the quotient 1; wherefore every single repetend is equal to a vulgar frac tion, the numerator of which is the repeating figure, and the denominator 9. = .001; whence =33 002, = 999 999 003; and the same holds true universally. Wherefore every pure repetend is equal to a vulgar fraction, the numerator of which consists of the repeating figures, and its denominator of as many nines as there are repeating figures; which was to be shewn. -250. When any part of the repetend is a whole number. RULE. Subjoin as many ciphers to the numerator as the highest place of the repetend is distant from the decimal mark *. 6. Reduce i.oi and 12.78 to fractions. 7. Reduce 2.46, is.24, and 1234.8 to vulgar fractions. An S. Reduce 1.3, 21.7, and 312.4 to equal vulgar fractions. 251. To reduce a mixed repetend to its equivalent vulgar fraction. RULE I. Prefix as many nines as there are places in the repetend, to as many ciphers as there are places in the finite part, for a denominator. II. From the given mixed repetend subtract the finite part for a numerator, and reduce the fraction to its lowest terms for the answer. k This rule may be explained by example 6; where if we suppose ioi to be wholly a decimal, its equivalent vulgar fraction will be by the preceding 101 999 rule; but i.oi is ten times ioi, whence the foregoing fraction multiplied by 101 999 X 10,) or will be the value of i.oi. Again, if i278 10, (thus 1010 be considered as a decimal, its equivalent vulgar fraction will be will be 100 times as great as that equal to the latter, that is, 12.78 = which is the rule. i2.78 is 100 times .i278; wherefore the vulgar fraction, equal to the former, 127800 9999 10. Reduce .5925 to an equivalent vulgar fraction. Thus 9990 denominator. and 5925 55920 numerator. 5 18 11. Reduce .27 and .53 to vulgar fractions. Ans. and 12. Reduce .345, .1234, and .43210 to vulgar fractions. 8 15 252. To make dissimilar repetends similar and conterminous. RULE I. Consider which of the given repetends begins the farthest from unity, and continue each of the other repetends to as many places from unity, putting a dot over the right hand figure of each; this will make them similar. II. Continue all the repetends to as many more figures as are equal to the least common multiple of the several numbers of places in all the repetends, and place a dot over the last, or right hand figure; this will make the repetends conterminous. 13. Given the following dissimilar repetends .321, 5.47, 3.2, .123, and 2.39, to make them similar and conterminous. |