XII. Circulating decimals, with a simple repetend, are changed to common fractions by using the repetend as numerator, and as many 9s as there are figures in the repetend as denomi nator. 5. Change 3, §, and g, severally, to circulating decimals, without any formal division, and prove by rechange to common fractions by the 12th principle above. 6. Change, and 3, severally to circulating decimals by inspection, and prove by rechange to common fractions. 25 768 7. Changes, .7., 829., 184, 88, severally to circulating decimals by inspection, and prove by rechange to common fractions. 2. Decimal Fractions with Compound Repetends. 1. Change to the form of a decimal fraction, and prove by rechange to its original form. ¿=·16=√b+9%=3%+3%=£5=£• Suggestive Questions.-Of what denomination is the 1 in the decimal fraction? Of what denomination would the 6 with a dot over it have been, had it stood in the place of the 1? What, then, is its denomination standing one rank more to the right? What prime factor in the denominator of causes it to repeat? Why are the two parts of the decimal changed separately to common fractions? 2. Change to a decimal fraction, first determining what figures, if any, in the denominator will cause it to repeat, and prove by rechange to its original form. 6 Suggestive Questions.-Why is the 6 changed to in place of § ? What prime factor in causes the decimal to repeat? From the last two demonstrations, then, may it not be legitimately inferred as a principle in arithmetic, that, XIII. Circulating decimals, with compound repetends, may be changed to common fractions, by changing separately the repetends and the figures that precede them to common fractions, and then adding them together. 3. Change the following common fractions to a decimal • form, first determining what prime factors in the denominators, if any, will cause them to repeat. Prove by a rechange to their original form: 1, 15, 18, 22, 26. d. To change a mixed number to an equivalent improper fraction. 1. Change 4 to an improper fraction, and prove by rechange to its original form. See definition 1 to this section, p. 194. 2. Change 9, 624, 81§, 67, severally to improper fractions, and prove by rechange to their original form. 3. Change 14, 279, 38, and 84, severally to improper fractions, and prove by rechange to their original form. e. To change a compound fraction to an equivalent simple one, or to a whole or a mixed number. 1. Change of 4 of 18 to an equivalent simple fraction. 3.4.10 120 2 Suggestive Questions.-What is the meaning of the word of in common fractions? What is indicated by the period placed between numbers? What, then, is the process for simplifying compound fractions? Remark. By cancelling, that is, by striking out the prime factors common to both terms in the compound fraction, the intermediate multiplication is always shortened, and sometimes, as in the present instance, rendered wholly unnecessary. For, if 10 in the numerator is mentally resolved into its two factors, 2 and 5, it will be perceived that the 3 and 5 in the numerator will cancel the 15 in the denominator; and, as the two 4s destroy each other, nothing remains but 2 and 5, making . 2. Change of of of 1⁄2 to an equivalent whole number, cancelling by inspection. Ans. 1. 3. Change of 2 of 16 of to an equivalent simple fraction, cancelling by inspection. 4. Change of of of to an equivalent mixed number, cancelling by inspection. Ans. 4. 5. Change of of % of to an equivalent simple frac tion, by cancelling two figures in the numerator and one in the denominator. Ans. 6. Change of of to an equivalent simple fraction. Ans. 14. f. To change fractions of different denominators to equivalent fractions with a common denominator. 1. Change and to equivalent fractions with a common denominator. See Oral Arithmetic, Chap. III., Sect. III., p. 89. 2. Change and to equivalent fractions with a common denominator. 3. Change and to equivalent fractions with a common denominator. 1st. By multiplication; 2d, by division; in both cases by inspection. Whenever an operation can be performed by division as well as by multiplication, the former should always be chosen, since it leaves the result in a more simple form. 4. Change and to equivalent fractions with a common denominator by inspection. Can this be done by division? Why? Must both fractions or only one be changed, in order to bring them to the same denomination? 5. Change and 5 by inspection to equivalent fractions with a common denominator. 1st, By multiplication; 2d, by division. 6. Change and to equivalent fractions with a common denominator by inspection. By division and multiplication. g. To change fractions of different denominators to equivalent fractions with the least common denominator. 1. Change and to equivalent fractions with the least common denominator, by inspection. (The factor 2 in the second denominator may be omitted, as it is to be found in the first. See Oral Arithmetic, Chap. III., Sect. III., p. 89.) Prove by changing each fraction to its lowest denomination. 2. Change and to equivalent fractions with least common denominator. What factors may be omitted in 12? Why? Prove as in last example. 3. Change,,,,, to equivalent fractions with least common denominator by inspection. The 2d, 3d, 5th, and 6th denominators may be omitted. Why? Point them out in the 1st and 4th. Prove, as in last example. Remark. Hitherto fractions have been changed to equivalent fractions with least common denominator by inspection merely. When the denominators are large and numerous, however, this is somewhat difficult for the unpractised student. It may be proper, therefore, to show how such changes may be effected by calculation on the slate or black-board, as follows: 4. Change, 1, 1, and, to equivalent fractions with the least common denominator. Prove as in last example. By analyzing the denominators into their prime factors, we have Suggestive Questions.-Are all the underlined factors to be found in the denominators of the fractions marked a and b ? Should they be omitted, then, in finding the lowest common denominator? What is the product of the factors that are not underlined? (80.3.5.) Has this product every factor contained in all the given denominators? Will it form their common denominator, then? Does it contain no more factors than they do? Will it form, then, their lowest common denominator? If, then, the fraction a is to be changed to the denomination of 1200, and one of its factors (80) is given, how shall the other factor be found? Should the second factor for each of the other denominators be found by the same process (division)? Probably, then, the numbers cannot be arranged more conveniently than as follows: 5. Change 1, 4, 7, and §, to least common denominator. Prove as by last example. 3.5 2.2 2.2.3 2.2.2=120. 6. Change, 2, 14, by calculation, to equivalent fractions with least common denominator. Prove by changing each fraction severally to its lowest denomination. 7. Change,,, and, by calculation, to equivalent fractions with least common denominator. Prove as in last example. 8. Change,,, and §, by calculation, to equivalent fractions with least common denominator. Prove as in last example. 2 9. Change,, 1, 3, 15, and 35, by calculation, to equivalent fractions with least common denominator. Prove as in last example. 10. Change, §, 3, 4, 1, by inspection, to equivalent fractions with least common denominator. One only of these denominators is a factor in the least common denominator. Which is it? Prove as in last example. 8 27' 11. Change,,,, 2, 4, by inspection, to equivalent fractions with least common denominator. Prove as in last example. 12. Change,, 11, 11, 71, 385, by calculation, to equivalent fractions with least common denominator. Prove as in last example. 5 13. Change, 1, §, 1, 14, by calculation, to equivalent fractions with least common denominator. Prove as by last example. 14. Repeat examples 4, 5, 6, 7, 8, 9, 12, and 13, by inspection, and prove as above. h. To change a complex fraction to an equivalent simple one. 1. Change to an equivalent simple fraction; in other words, perform partially the division indicated. 2. Change to an equivalent simple fraction in its lowest expression, performing both changes at one operation by inspection. Ans. 14 |