7. What is the value of of 15% of 5% of 100? 8. What is of of 1? Ans. 575813. 9. What is the value of of of of $72? Ans. $1.75. 10. What is the value of of 7 of 1 of 33 gallons? of 3 of ? Ans. gal. 11. What part of a ship is Ans. $6.75. A COMMON DENOMINATOR. 224. Fractions have a common denominator when all their denominators are alike. 225. A common denominator of two or more fractions is a common multiple of their denominators; and their least common denominator is the least common multiple of their denomi nators. 226. To reduce fractions to a common denominator. Ex. 1. Reduce,, and 1 to other fractions of equal value, having a common denominator. 8 X 12 X 16 = 1 5 3 6 common denominator. Ans. We first multiply the numerator of by the denominators 12 and 16, and obtain 1344 for a new numerator. We next multiply the numerator of by the denominators 8 and 16, and obtain 640 for a new numerator; and then we multiply the numerator of by the denominators 8 and 12, and obtain 1056 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 217.) 16 X 3 = tor. 16 3 X 11=33, “. 48, least common multiple, and least common denomina Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of this number, 48, as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for, we take 7 of 48, the least common denominator, by dividing it by 8, and multiplying the quotient 6 by 7. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as both terms are multiplied by the same number. (Art. 217.) The method used in the second operation, it will be perceived, expresses the fractions of the result in lower terms than that used in the first. On this account it is often to be preferred to the other. RULE. - Find the least common multiple of the denominators for the LEAST COMMON denominator. Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. Or, Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for A COMMON denominator. NOTE 1. - Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, before finding a common denominator, and all to their lowest terms, before finding the least common denominator. NOTE 2. - Fractions may sometimes be reduced to a common denominator most readily by multiplying both terms of one or more of them by such a number as will make all the denominators alike. Thus and may be brought to a common denominator simply by multiplying both terms of the by 2, and changing in that way its form to 2. NOTE 3.- Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. EXAMPLES. Reduce the following fractions to their least common denom inator: 2. Reduce 2, 3, 1, and 12. 6 18, Ans. $4, 42, 93, 33. 704 3. Reduce fr,,, and . Ans. 1%, 7, 128, 1720. 16. Reduce,, and to fractions having a common de nominator. 80 96 Ans. 120, 120, 125. 360 560 Ans. 98, 98, 183. Ans. 546 572 616 100T 10019 1001• Ans. 185 85 1020 612 2295 2295, 2295. Ans. 2940 540 26316 17. Reduce, §, and . 27. Reduce 29, 121, and 1720 227. 2295 22959 2295 Ans. 1, 2, 4, 336. Ans. 528 756 6039 1188 T1889 1188 8008 7007 14014, 14014 722813 1784315529 1784315529 178431552 ADDITION OF COMMON FRACTIONS. Addition of fractions is the process of finding the value of two or more fractions in one sum. NOTE. Only units of the same kind, whether integral or fractional, can be collected into one sum; if, therefore, the fractions to be added do not express the same fractional unit, they require to be brought to the same, by being reduced to a common or the least common denominator. 228. To add together two or more fractions. Ex. 1. Add 12, 12, 12, and 11 together. Ans. 2 These fractions 24. all being twelfths, that is, having 12 Thus we for a common denominator, we add their numerators together, and write their sum, 26, over the common denominator, 12. obtain, which, being reduced, 21, the sum required. = 2. What is the sum of 1, 2, 11, and 13? 8 12 OPERATION. least common denominator. 7 = 210 16/20 5 = tors. The given fractions not expressing the same kind of fractional unit, we reduce them to their least common denominator, and thus make the fractional parts all of the same kind. The fractions now all expressing two-hundred-fortieths, we add their numerators, and write the result, 631, over the least common denominator, 240, and obtain 211, the answer required. RULE. Reduce the fractions, if necessary, to a common, or the least common denominator, and write the sum of the numerators over their common denominator. NOTE 1.- Mixed numbers must be reduced to improper fractions, and compound fractions to simple fractions, and each fraction to its lowest terms, before attempting to obtain the common denominator. NOTE 2. In adding mixed numbers, the fractional parts may be added separately, and their sum added to the amount of the whole numbers. EXAMPLES. 3. Add 17, 17, 17, 14, 14, and 14 together. 5. What is the sum of 47, 17, 19, and 77? 4. Add 23, 23, 11, 13, and 13 together. 19 6. What is the sum of,,, and 10? Ans. 17. 7. What is the sum of £11, 31, §§7, and Ans. 319. 1? Ans. 243. 19. Add 63, 78, and 43 together. Ans. 17. 252 Ans. 9115. Ans. 183 17? Ans. 40. 22. What is the sum of 37111, 61418, and 812 ? Ans. 106837. 23. Add of 18, and 1 of 3 of 6 together. 24. Add & of 18, and of 1 of 7 Ans. 12883 together. 229. To add any two fractions, whose numerators are alike. product, which is 20; and the 9 being written as a numerator of a fraction, and the 20 as its denominator, the result,, is the answer required. The reason of the operation is, that the process reduces the fractions to a common denominator, and then adds their numerators. Hence, to add two fractions whose numerators are a unit, Write the sum of the given denominators over their product. 2. Add 2 to 3. Sum of the denominators X OPERATION. by one of the numerators, (4 + 5) × 3 Product of the denominators, 4 x 5 Ans. 17. By multiplying the sum of the denominators by one of the numer ators for a new numerator, and the denominators together for a new denominator, we reduce the fractions to a common denominator, and add their numerators, and thus obtain 27 12, the answer required. Hence, to add fractions whose numerators are alike, and greater than a unit, = Write the product of the sum of the given denominators by one of the numerators over the product of the denominators. |