Examples. 1. Required the least equivalent common fraction of the mixed repetend, 2.4`18'. Now, 18 2.4'18' 2+ + '18′ = 2 + 1 + 99% = 233. Ans. = 4 0 2. Required the least equivalent common fraction of the mixed repetend .5'925'. We have, 5'925'10 +0 = 925 9990 16 27 3. What is the least equivalent common fraction of the repetend .008'497133' ? 4. Required the least equivalent common fractions of the mixed repetends .138, 7.5'43', .04'354', 37.5'4, .6'75', and .7'54347'. 5. Required the least equivalent common fractions of the mixed repetends 0.7'5, 0.4'38', .09`3, 4.7`543', .009`87', and .4`5 CASE III. 222. To find the finite figures and the repetends correspondIng to any common fraction. 1. Find the finite figures and the repetend corresponding to the fraction 580. 8d. The number of finite decimals preceding the first figure of the repetend will be equa. to the greatest number of factors 2 or 5: in this example it is 3. 4th. When a remainder is found which is the same as a previous dividend, the second repetend begins. 5th. The number of figures in any repetend will never exceed the number, less 1, of the units in that factor of the denominator which does not contain 2 or 5. In the example, that number is 7 and the number of figures of the repetend. is 6. Rule.-Divide the numerator of the common fraction, reduced to its lowest terms, by the denominator, and point off in the quotient the finite decimals, if any, and the rep-tend. Examples. 1. Find whether the decimal, equivalent to the common fraction is finite or repeating required the finite figures, if any, and the repetend. 249 29304' 83 9768 249 OPERATION. 83 = 29304 9768 83 = 2×2×2×1221 ANALYSIS.-We first reduce the fraction to its lowest terms, giving 83 We then search for the factors 2 and 5 in the denominator, and find that 2 is a factor 3 times; hence, we know that there are three finite decimals preceding the repetend. We next divide the numerator 83 by the denominator 9768, and note that the repetend begins at the fourth place. After the ninth division, we find the remainder 83; at this point the figures of the quotient begin to repeat; hence, the repetend has 6 places. 9768)83.00 ... (.008'497133′ 2. Find the finite decimals, if any, and the repetend, if any, of the fraction. 3. Find the finite decimals, if any, and the repetend, if any, of the fraction 1160. 4. Find the finite decimals, if any, and the repetend, if any, of the fractions 12 80 72 123 1351 135 223. Properties of the Repetends. There are some properties of repetends which it is important to remark. 1. Any finite decimal may De considered as a repeating decimal by making ciphers recur; thus, .35 .350 .35'00' =.35'000' = .35'0000', &c. 2. Any repeating decimal, whatever its number of figures, may be changed to one having twice or thrice that number of figures, or any multiple of that number. Thus, a repetend 2.3'57' having two figures, may be changed to one having 4, 6, 8, or 10 places of figures. For, 2.3'57' = 2.3'5757′ =2.3'575757′ = 2.3'57575757', &c.; so, the repetend 4.16'316' may be written 4.16'316' 4.16'316316' 4.16'316316316', &c.; and the same may be shown of any other. Hence, two or more repetends, having a different number of places in each, may be reduced to repetends having the same number of places, in the following manner: Find the least common multiple of the number of places in each repetend, and reduce each repetend to such number of places. 3. Any repeating decimal may be transformed into another having finite decimals and a repetend of the same number of figures as the first. Thus, .'57' = .5'75' = .57'57′ = .575'75′ = .5757`57′; and 3.4'785′ = 3.47`857′ = 3.478'578′ = 3.4785'785′; and hence, any two repetends may be made similar. 222. How do you find the finite figures and the repetend corro sponding to any common fraction?-223. 1. How may a finite decimal be made a repeating decimal? 2. When a repetend has a given number of places, to what other form may it be reduced? How b. Into what form may any repeating decimal be transformed? These properties may be proved changing the repetends into their equivalent common fractions. 4. Having made two or more repetends similar by the last article, they may be rendered conterminous by the previous one; thus, two or more repetends may always be made similar and conterminous. 5 If two or more repeating decimals, having several repetends of equal places, be added together, their sum will have a repetend of the same number of places; for, every two sets of repetends will give the same sum. 6. If any repeating decimal be multiplied by any number, the product will be a repeating decimal having the same number of places in the repetend; for, each repetend will be taken the same number of times, and consequently must produce the same product. Examples. 1. Reduce .13'8, 7.5'43' .04'354', to repetends having the same number of places. Since the number of places are now 1, 2, and 3, the least common multiple is 6, and hence each new repetend will contain 6 places; that is, .138.13888888'; 7.5'43' = 7.5'434343'; and .04'354' = .04'354354'. 2. Reduce 2.4'18', .5'925', .008'497133', to repetends having the same number of places. 3. Reduce the repeating decimals 165.'164', .'04', .037 to such as are similar and conterminous. 4. Reduce the repeating decimals .53, .475', and 1.757' to such as are similar and conterminous. 223. 4. To what form may two or more repetends be reduced? ADDITION. 224. To add repeating decimals. I. Make the repetends, in each number to be added, similar and conterminous : II. Write the places of the same unit value in the same column, and so many figures of the second repetend in each as shall indicate with certainty, how many are to be carried from one repetend to the other: then add as in whole numbers NOTE.-If all the figures of a repetend are 9's, omit them and add 1 to the figure next at the left. Examples. 1. Add .125, 4.163', 1.7143', and 2.54', together. 4371 5454 1.7143' 1.71'4371′ = 1.71'437143714371'. 2.54'2.54'54' =2.54 545454545454' The true sum = 8.54'854470131697' 1 to carry. 2. Add 67.3'45', 9.'651', .25', 17.47, 5, together. SUBTRACTION. 225. To subtract one repeating decimal from another. ! Make the repetends similar and conterminous : II. Subtract as in finite decimals, observing that when the repetend of the lower line is the larger, 1 must be carried to the first right-hand figure. 224. How do you add repeating decimals?-225. How do you sub tract repeating decimals. y* |