2. Reduce 2, 4, and to equivalent fractions having a com mon denominator. . Ans. 198, and f 1409 44. Ans. 11, 18, and 389. 5049 68 350 Ans. 18, 38, 188, and 18. 8 3. Reduce, 2, and 3. 4. Reduce 3, 3, and 4. 5. Reduce,,, and . 6. Reduce,, and . 7. Reduce, and §. 8. Reduce,, and 9. Reduce 3, 3, and 45. 10. Reducer, f, and. (a) The foregoing rule will always give a common denominator, but not always the least integral common denominator; this, however, may always be effected by 13. Reduce , 1, 1, and 1§. 14. Reduce, 27, 4, and • 15. Reduce, 17, 5, and 1's. RULE 2. Reduce each fraction, if necessary, to its lowest terms (Art. 141). Find the least common multiple of the denominators (Art. 127) for a common denominator. Divide this multiple by each given denominator, and multiply the several quotients by the respective numerators for new numerators. NOTE 1. Each of these rules is founded on the principle that multiplying both terms of a fraction by the same number does not alter its value. 16. Reduce, 8, and 7. 147. Rule for finding the least common denominator? Rule for finding the Gumerators? Principle? NOTE 2. The first clause of Rule 2 is omitted by many authors, but its necessity is apparent from the following example: 8 20. Reduce §,, and to equivalent fractions having the least common denominator. Disregarding the first clause of the rule, we find 72 to be the least common multiple of the denominators, and the fractions §, , and, reduce to 4, 3, and 43; but, regarding the first clause, we have §,, and }, }, and } = 12, 12, and 12 which have a common denominator less than 72. NOTE 3. Ans. 18, 18, 18, and 18. In this and the following problems, each fraction should be in its simplest form before applying the rule. REMARK. The numerators, as well as the denominators, of fractions, may be made alike by reduction; thus, and are equal in value to 1 and 1; also ‡ and f = {} and 12; also 4, 1, and 21, 24, and 14; etc. The process is simple, but of little practical importance, and therefore seldom presented in Arithmetic. 147. May the numerators of fractions be made alike? How? PROBLEM 10. 148. To reduce a fraction of a higher denomination to a fraction of a lower denomination. Ex. 1. Reduce of a penny, to the fraction of a farthing. As 1 penny is equal to 4 farthings, so any fraction of a penny will be 4 times as great a fraction of a farthing; .. d. 4 times fqr. = #qr.. Ans. 2. Reduce of a shilling to the fraction of a farthing. As 1s. is equal to 12d., so 48. = 12 times d. 4d., and 4d. 4 times 4qr. qr., Ans. Hence, RULE. Multiply the fraction by such numbers as are necessary to reduce the given to the required denomination. 3. Reduces. to the fraction of a farthing. 3s. (=3d. X 12)=3d. (=3qr. × 4) = 28qr., Ans. ; or, 7 × 12 × 4 28 7 X 12 X 4 36 36 3 = qr., Ans., as before. 3 NOTE 1. The sign of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 3, no sign is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from canceling 20 in 240, and the 3 from canceling 4 in 12. 4. Reduce of a ton to the fraction of a dram. 7 × 20 × 4 × 25 × 16 X 16 240 44800 dr., Ans. NOTE 2. In the first statement of Ex. 5, the 16, in the numerator, is equal to 33, and, in the second statement, the 33 is retained in the numera tor as a factor in the dividend, and the 2 is put in the denominator as a fac tor in the divisor. 148. Rule for reducing a fraction from a higher to a lower denomination? Explanation? How is Ex. 5 solved? 6. Reduce of a pound, Troy Weight, to the fraction of a grain. Ans. 18. 7. Reduce of a pound, Apothecaries' Weight, to the frac tion of a grain. 800 Ans. 188. 8. Reduce of a day to the fraction of a second. Ans. 144. 9. Reduce of a bushel to the fraction of a pint. 15. Reduce 33 of an acre to the fraction of a square yard. 16. Reduce yd. of cloth to the fraction of an inch. 17. Reduce circ. to the fraction of a second. of a ton to the fraction of an ounce. 18. Reduce 20. Reduce of a day to the fraction of a second. £ to the fraction of a farthing. 21. Reduce of a bushel to the fraction of a pint. PROBLEM 11. 149. To reduce a fraction of a lower denomination to a fraction of a higher denomination. Ex. 1. Reduce of a barleycorn to the fraction of an inch. In 15 barleycorns there is only of 15 inches, so in of a 3 barleycorn there is only of of an inch of an inch, Ans. 2. Reduce of a gill to the fraction of a quart. As 1 gill is of a pint, so gi. is of pt. pt. and, for ▲ like reason,pt. isofqt.qt., Ans. Hence, = = RULE. Divide the given fraction by such numbers as are required to reduce the given to the required denomination. 149. Rule for reducing a fraction from a lower to a higher denomination' Explanation? 3. Reduce 2qr. to the fraction of a shilling. ÷ ÷ 28qr. (=2&d. 4) = 3d. (= 3s. 12)=3s., Ans.; or, 4. Reduce 448oodr. to the fraction of a ton. = 7 3 × 16 × 16 × 25 × 4 × 20 240 5. Reduce 122b.c. to the fraction of a rod. 1980 7 X 3 X 12 X3 X5 10 tons, Ans. 6. Reduce 188gr. to the fraction of a pound, Apothecaries' Weight. Ans. 1200 7. Reduce 18gr. to the fraction of a pound, Troy Weight. 9. Reduce in. to the fraction of a yard, Cloth Measure. PROBLEM 12. Ans. 150. To reduce a fraction of a higher denomination to whole numbers of lower denominations. Ex. 1. Reduce to shillings and pence. Ans. 3s. 4d. £ (s. 20) = s. 33s.; again 3s. (=}d. × 12)= X 4d.;.. £ 3s. 4d., Ans. £= Hence, RULE. Reduce the given fraction to a fraction of the next lower denomination (Art. 148); then, if the fraction is improper, reduce it to a whole or mixed number (Art. 140). If the result is 150. Rule for reducing a fraction of a higher denomination to integers of lower denominations? Explanation? |