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.
(7) 221, 8, 4211, 207.

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,
divide numerator and denominator by any common
factors; this process is called cancelling. If the
common factors are not recognizable, divide top and
bottom by the G.C.M.

Thus ===2}.

EXAMPLES. (a) 134, (b) 858, (c) 113, (d)

72
1869

8049

(e) 1%, (f) 418, (g) 224,

43118, (4) 1173,

22176
233287

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
35
250 896

6

12, (i) 38072, (j)
(n) 27, (0) 631.

42

801

(d) 3, (e) 121,
(j) H, (k) §, (1)

(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
384

Ans. 55

Ans. 307

34 50

36549

5436
1836

Ans. 1242,

307

2484

2039

969

201

3279

1535

621

2182
1039 471 1399

(7)

3672
1236)

638,

3612

138 24 1236'

RULE V. To convert fractions with different
denominators into fractions having a common denomi-
nator. Find the L.C.M. of the denominators, this
will give the common denominator. Convert,,,
, to a fraction having the least common denominator.
L.C.M. of 2, 3, 5, 9, is 90.

Examples of this will be required in addition and
subtraction of Vulgar Fractions.

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