Find L.C.M. of (a) 2, 3, 4, 5, 6, 7, 8; (b) 2, 4, 6, 8, 10, 12; (c) 3, 5, 7, 9, 11; (d) 3, 8, 24, 4, 6; (e) 4, 8, 12; (ƒ) 3, 9, 27, 54; (g) 12, 15, 18, 9; (h) 72, 4, 9, 36; (i) 6, 11, 7, 4; (j) 3, 9, 4, 12; () 16, 4, 32, 11; (1) 7, 14, 11, 88; (m) 44, 11, 6, 7; (n) 42, 12, 6, 7; (0) 84, 9, 8, 7; (p) 18, 9, 7, 3, 14; (g) 29, 7, 28, 58. Ans. (a) 840, (f) 54, (g) 180, (1) 616, (m) 924, (b) 120, (c) 3465, (d) 24, (e) 24, (h) 72, (i) 924, (j) 36, (k) 352, (n) 84, (0) 2268, (p) 126, (q) 812. (r) What is the least number of weeks in which a man can earn an exact number of sovereigns whose wage is 17s. per week? Ans. 20 weeks. (s) What is the smallest sum that could be paid with half-crowns, threepenny pieces, and fourpenny pieces respectively? Ans. 5s. Find the L.C.M. of-(1) 8, 16, 24, 3, 9, 7, 63, 18, and 4; (2) 14, 7, 21, 16, 8, 24, 35, 60; (3) 36, 2, 18, 4, 9, 3, 27, 28; (4) 2, 3, 8, 7, 4, 16, 12, 14; (5) 12, 3, 4, 5, 60, 22, 44; (6) 3, 13, 11, 2, 4, 6, 8, 22; (7) 2, 3, 9, 5, 45, 18, 27, 16; (8) 22, 3, 11, 66, 24, 18, 9; (9) 11, 33, 9, 27, 44, 28; (10) 32, 8, 4, 0, 1, 30, 16; (11) 3, 7, 2, 16, 8, 14, 24, 36; (12) 9, 8, 4, 7, 28, 36, 16, 18, 14. Ans. (1) 3024, (2) 3360, (3) 756, (4).336, (5) 660, (6) 3432, (7) 2160, (8) 792, (9) 8316, (10) 480, (11) 1008, (12) 1008. VULGAR FRACTIONS. A Fraction is a part of a whole number. A Vulgar Fraction consists of a Numerator and Denominator, of which the denominator or lower number represents the number of parts into which the whole is divided; and the numerator the number of these taken, as . A proper fraction has its denominator greater than its numerator, as ; and an improper fraction has the denominator less, as . A complex fraction is a fraction of a fraction, as of 1. A compound fraction has the numerator or denomi1 2 3 3호 321 nator, or both, in the form of fractions, as A mixed number consists partly of an integer or integers, and a fraction, as 7. RULE I. To reduce a mixed number to an improper fraction. Bring the integers to the parts designated by the denominator, and add the fractional parts. EXAMPLES. (a) 71, (b) 82, (c) 31, (d) 214, (e) 815, (ƒ) 715; (g) 3621, (h) 7417, (i) 841, (j) 31, (k) 61, (1) 821%, (m) 6}, (n) 71. Ans. (a) 640, (b) 493, (c) 10, (d) 151, 9 6 (f) 106, (g) 393, (h) 1269, (i) 1597, 15 17 500, (1) 1485, (m) 10, (n) 31. 13 18 4. 19 7 (e) 153, (j) 348, (k) 11 Reduce the following mixed numbers to improper fractions: (1) 301, 4517, 4613. Ans. 856, 28 8 27 18 13%, 23, 441, 291. Ans. 15, 331 (3) 651, 61, 57, 5014. Ans. 857, 18 94, 854 7 18, 19, Ans. 872 25 4 20 3 873 24 Ans. 367 (6) 331, 30, 283, 261. 367. 4 14 11 Ans. 167. (8) 21%, 131. Ans. 131, 79 (9) 13%, 72, 958, 4615. Ans. 81, 794 (10) 122%, 40%, 76, 71. Ans. 856, 854 (11) 136, 45%, 7811, 881. 6 861 9 367 9 Ans. 817 (12) 39, 801, 33, 3043. 914. (13) 81, 74, 64. 16 316 368 3 RULE II. To reduce an improper fraction to a mixed number. This is the converse of the preceding rule. Divide the numerator by the denominator; thus 20=63. For practice the student can convert the answers of Rule I. to mixed numbers, the answers to which will be the exercises in that rule. Reduce the following improper fractions to mixed or whole numbers : RULE III. To multiply a fraction by a whole number, multiply the numerator only by that number, and to divide a fraction by a whole number, divide the numerator only by that number, if it will exactly contain it, and if not, multiply the denominator only by the divisor. Thus x4 =2, and ÷2=, or = 3. 4 3×2 = = RULE IV. To reduce a fraction to its lowest terms, Thus ===2}. EXAMPLES. (a) 134, (b) 858, (c) 113, (d) 72 8049 (e) 1%, (f) 418, (g) 224, 43118, (4) 1173, 22176 21787 9361 (h) 819 9547 (1) 1+21, (m) 678, 44229 (b) 107, (c) 1, 91 (h) 10%, (i) 32, 1994, (m) 338, (n) 27, (0) 55 Reduce to their lowest terms- 60 6 12, (i) 38072, (j) 42 801 (d) 3, (e) 121, (1) 80, 85, 1987. Ans. 25, 195, 286. (2) $38, 128, 6372 104, 74. (3) 9189 6 3 8 2 Ans. 71 401 16649 148 1049 4 16 2496 20736 2052102601 27648 28809 6589 276 Ans. 55 Ans. 307 34 50 36549 5436 Ans. 1242, 307 2484 2039 969 201 3279 1535 621 2182 (7) 3672 638, 3612 138 24 1236' RULE V. To convert fractions with different Examples of this will be required in addition and Reduce to a common denominator- RULE. Turn the fractions into equivalent fractions with the least common denominator, and add their numerators together. 3 EXAMPLES. (a) + + + + }, (b) § + 1 + 1, (c) ÷ + 22 + 1/1, (α) } + { ++ }, +3, (9) + 1 + 18+ 3, (2) 11 + 3 + 4 + 8, (i) + 8 +3, (j) 1's +++ (k) + 4 +36 (1) +2/2+355 (m) 1+2+3+3, (n) 13+13+ 212 213 + 2 1/2 (0) 1+11+14, (p) 21++3+1, (q) 3+7+ 1, (r) 2+3+ (e) { + } + ÷ + }, (ƒ) } + 1 + 8409 9 19 59 Ans. (a) 21, (b) 8, (c) 11, (d) 538, (e) 35, (f) 3 (g) 1137, (h) 413, (i) 164, (j) 1,859 59 1329 53 305 (k) 115, (1) 58, (m) 553, (n) 5,565, (0) 21 277 (p) 93, (q) 115, (r) 61⁄21⁄2. 10929 1386' (1) 3++++7. Ans. 2,923 (2) 1+1+ +++ Ans. 217. (3) +3 +3 +3 +3 +3. Ans. 2308 (4) ++++++. Ans. 3. +++++22. Ans. 2. (6) + (6)+1 Ans. 3. (7) { + } + { + } + ↓ +†. 글+++++. (5) ++++ 420 13 3 Ans. 2118 (8) √ + 3 3 + 3 + 1+. Ans. 27 (9) 콩++++우. Ane. 2골을뜸. (10) ++++ 11. Ans. 2. (11) ++&++fs+. Ans. C |