330. PROP. VI.- Diminishing the numerator and denominator by the same fractional part of each does not change the value of a fraction. Be particular to master the following, as the reduction of circulating decimals to common fractions depends upon this proposition. 1. The truth of the proposition may be shown thus: 9 9 – of 9 9 – 3 6 3 12 – 12 – 4 of 12 = 12 - 4 = 4 Observe that to diminish the numerator and denominator each by of itself is the same as multiplying each by ş. But to multiply each by , we multiply each by 2 (235—II), and then divide each by 3 (235—III), which does not change the value of the fraction; hence the truth of the proposition. 2. From this proposition it follows that the value of a fraction is not changed by subtracting 1 from the denominator and the fraction itself from the numerator. . 3 3 3 2 3 3 Thus, 5 = 5-1-4 Observe that 1 is the # of the denominator 5, and is of the numerator 3; hence, the numerator and denominator being each diminished by the same fractional part, the value of the fraction is not changed. DEFINITIONS. 331. A Simple Decimal is a decimal whose numerator is a whole number; thus, 1903 or .93. Simple decimals are also called Finite Decimals. 332. A Complex Decimal is a decimal whose numerator is a mixed number ; as 20 or.265. There are two classes of complex decimals : 1. Those whose value can be expressed as a simple decimal (326), as .231 = .235 ; .323 = .3275. 2. Those whose value cannot be expressed as a simple decimal (327), as .53 = .53333 and so on, leaving, however far we may carry the decimal places, į of 1 of the lowest order unexpressed. See (328). 333. A Circulating Decimal is an approximate value for a complex decimal which cannot be reduced to a simple decimal. Thus, .666 is an approximate value for .666 (329). 334. A Repetend is the figure or set of figures that are repeated in a circulating decimal. 335. A Circulating Decimal is expressed by writing the repetend once. When the repetend consists of one figure, a point is placed over it; when of more than one figure, points are placed over the first and last figures; thus, .333 and so on, and .592592 + are written .3 and .592. 336. A Pure Circulating Decimal is one which commences with a repetend, as .8 or .394. 337. A Mixed Circulating Decimal is one in which the repetend is preceded by one or more decimal places, called the finite part of the decimal, as.73 or .004725, in which .7 or .004 is the finite part. ILLUSTRATION OF PROCESS. 338. PROB. I.-To reduce a common fraction to a decimal. Reduce f to a decimal. EXPLANATION. – 1. We annex 5 = 8000 – 1000 = .375 the same number of ciphers to both terins of the fraction (235– Prin. II), and divide the resulting terms by 8, the significant figure in the denominator which must give a decimal denominator. Hence, expressed decimally is .375. 2. In case annexing ciphers does not make the numerator divisible (327) by the significant figures in the denominator, the number of places in the decimal can be extended indefinitely. In practice, we abbreviate the work by annexing the ciphers to the numerator only, and dividing by the denominator of the given fraction, pointing off as many decimal places in the result as there were ciphers annexed. Hence the following 339. RULE.—I. Annex ciphers to the numerator and divide by the denominator. II. Point off as many places in the result as there are ciphers annexed. EXERCISE FOR PRACTICE. 340. Reduce to simple decimals : 1. i 3. 1 5. y 4. 2. t. 4. 6. 7. 8. 16 Reduce to a complex decimal of four decimal places: 9. Pa 11. 12. 13. 15. Bolo 10. H 12. . 14. 16. 1: Find the repetend or approximate value of the following: 17. 1. 20. 1 23. 2 26. 85. 18. 3. 21. . 24. 27. 2475 19. 4. 22. 7. 25. 4. 28. 32 341. PROB. II.—To reduce a simple decimal to a common fraction. Reduce .35 to a common fraction. 35 y EXPLANATION.—We write the decimal with .00 = 100 = 20 the denominator, and reduce the fraction (255) to its lowest terms; hence the following 342. RULE.--Express the decimal by writing the denominator, then reduce the fraction to its lowest terms. EXAMPLES FOR PRACTICE. 343. Reduce to common fractions in their lowest terms: 1. .215. 4. .0054. %. .00096. 10. .0625. 2. .840. 5. .0125. 8. .008025. 11. .00512. 3. .750. 6. .0064. 9. .00075. 12. .00832. 344. PROB. III.-To find the true value of a pure circulating decimal. Find the true value of .72. ... 2 2 8 EXPLANATION.—In takv = 100 = 100 – 1= 99 =ū ing .72 as the approximate value of a given fraction, we have subtracted the given fraction from its own numerator, as shown in (329_V). Hence, to find the true value of no, we must, according to (330–VI, 2), subtract 1 from the denominator 100, which makes the denominator as many O’s as there are places in the repetend ; hence the following 345. RULE.— Write the figures in the repetend for the numerator of the fraction, and as many 9's as there are places in the repetend for the denominator, and reduce the fraction to its lowest terms. EXAMPLES FOR PRACTICE. 346. Find the true value of 1. 36. 4. 372. 7. i89. 10. 5368. 2. 18. 5. 856. 8. 324. 11. 2718. 3. 54. 6. i35. 9. 836. 12. 8163. Find the true value as improper fractions of 13. 37.8i. 16. 89.54. 19. 63.2745. 14. 9.108. 17. 53.324. 20. 29.1881. 15. 3.504. 18. 23.758. 21. 6.036. 347. PROB. IV.-To find the true value of a mixed circulating decimal. Find the true value of .3i8. (1) 018 = .0$i = 10 = 990 = 22 EXPLANATION.–1. We find, according to (344), the true value of the repetend .ois, which is .01). Annexing this to the .3, the finite part, we have .3;$, the true value of .3i8 in the form of a complex decimal. 2. We reduce the complex decimal .3%, or , to a simple fraction by multiplying, according to (300), both terms of the fractiou by 99, giving 10 = 311 = 's. Hence the true value of .3i8 is z's. (2) 318 Given decimal. ABBREVIATED SOLUTION.— Observe 3 Finite part. that in simplifying , we multiplied 315 311= both terms by 99. Instead of multi plying the 3 by 99, we may multiply by 100 and subtract 3 from the product. Hence we add the 18 to 300, and subtract 3 from the result, which gives us the true numerator. Hence the following 348. RULE.—I. Find the true value of the repetend, annex it to the finite part, and reduce the complex decimal thus formed to a simple fraction. To abbreviate the work: II. From the given decimal subtract the finite part for a numerator, and for a denominator write as many 9's as there are figures in the repetend, with as many ciphers annexed as there are figures in the finite part. EXAMPLES FOR PRACTICE. 349. Find the true value of 1. 72. 4. .04328. 7.000035739. 2. .959. 5. .00641. 8. .008302685. 3. .486. 6. .03289. 9. .020734821. |