To change Improper Fractions to Integers or to Mixed Numbers. WRITTEN WORK. 204. A fractional number, the numerator of which equals or exceeds the denominator, is called an improper fraction. 205. ILLUSTRATIVE EXAMPLE III. Change 12 and 17 as far as possible to integers. Explanation. — (1.) Since 12 twelfths make a (1.) 12) 60 unit, in 60 twelfths there are as many units as there are 12's in 60, which is 5. Ans. 5. (2.) In 17 there are as many units as there are 12’s in 47, which is 3 and 1. Ans. 3 14. (2.) 12) 47 206. The number 313 consists of an Ans. 311 integer and a fraction. A number consisting of an integer and a fraction is a mixed number. Ans. 5 207. Oral Exercises. a. Change to integral numbers: $; ; ; ; ; ; 44; 4; 4; 3; 4; 4; 7; ft; **; 19; 2; 18. b. Change to mixed numbers : ; ; ; ; ; *; 2; 3; 4; HS; 22; 4; it; 18; ? ; fe ; 37. c. Change to integers or to mixed numbers: 4; 4; $; 25; 5; 6; 34; it; 40; 198; p; . 208. From previous illustrations we may derive the following Rule. To change an improper fraction to an integer or a mixed number: Divide the numerator by the denominator. 209. Examples for the Slate. Change to integers or to mixed numbers: (13.) 14 (15.) 4 (17.) 1914. (19.) 497 days. (14.) 1934 (16.) 473 (18.) 7332 (20.) 4 years. To change an Integer or a Mixed Number to an Improper Fraction, 210. ILLUSTRATIVE EXAMPLE IV. Change 231 to fourths. WRITTEN WORK. Explanation. — Since in 1 there are 4 fourths, in 234 = Ans. 23 there are 23 times 4 fourths, or 92 fourths, 4 which, with 1 fourth added, are 93 fourths. 93 Ans, 8. 211. Oral Exercises. a. Change to improper fractions : 2}; 38; 23; 54; 24; 33; 6}; 53; 5}; 78; 7}; 8; 88; 9}; 98; 103. b. Change to improper fractions : 27; 28; 311; 34; 4}; 43; 5%; 93; 6; 7; 83; 94; 4; 43; 8}; 78. c. Change 5 to ninths; 11 to fifths ; 14 to thirds ; 8 to twelfths; 15 to fourths; 1 to sevenths. d. Among how many persons must 7 melons be divided that each may receive of a melon ? ? ? ? e. How many persons will 5} cords of wood supply if each person receives 4 of a cord ? of a cord ? $ of a cord ? 212. From previous illustrations may be derived the following Rule. To change an integer or a mixed number to an improper fraction: Multiply the integer by the denominator of the fraction, and to the product add the numerator; the result will be the numerator of the required fraction. 213. Examples for the Slate. Change the following to improper fractions : (21.) 69%. (24.) 7643 (27.) Change 48 to ninths. (22.) 2721. (25.) 10724. (25.) 1074 (28.) Change 567 to tenths. (23.) 10945. (26.) 663. (29.) Change 93 to forty-thirds. For other examples in reduction of fractions, see page 123. ADDITION OF FRACTIONS. To add Fractions having a Common Denominator. 214. ILLUSTRATIVE EXAMPLE I. Add f of an apple, of an apple, and of an apple. Ans. f of an apple. These fractions are like parts (eighths) of the same or similar units (apples). Such fractions are like fractions. 215. Like fractions have the same denominator, which, because it belongs to several fractions, is called a common denominator. a. Add 15, 1, and is. e. Add Po, do, and 44. b. Add 1 I&o, and ito. f. Add 1%, 48, and 45. c. Add %, 8, , and H. g. Add 767, 20%, and zooo: d. Add 18, is, t}, and Pg. h. Add 345, 352 , . How do you add fractions which have a common denominator ? To add Fractions not having a Common Denominator. WRITTEN WORK. X10 = 217. ILLUSTRATIVE EXAMPLE. Add &, , and 15. Explanation. To be added, these 2 x3x3 5 =90 1. c. denom. fractions must be changed to like frac tions, or to fractions having a common 8 = 6*16= 78 denominator. (Art. 215.) The new = 48 denominator must be some multiple of Yo = 7X = 3% the given denominators. A convenient Ans. 4% = 183. multiple is their least common multiple, which is 90. (Art. 182.) To change of to 90ths, the denominator 6 must be multiplied by 3x5, or 15; hence the numerator 5 must be multiplied by 15. (Art. 199.) Thus, á is found to equal 75. In a similar way will be found to equal 4%, and is to equal ff. Adding these fractions, we have $57, or 197, for the sum. 218. Oral Exercises. a. Add t, , and . Ans. Pa = 1: b. Add {, , and A. Ans. 38 = 237. c. Add }, &, and %. NOTE. When the denominators are prime to each other, the new denomi. nator will be the product of all the denominators, and the new numerators will be found by multiplying cach numerator by the product of all the denominators except its own. g. Add } and }; } and }; and }; and }; i and }; and ; } and to; } and } h. Add z and ; and ; f and %; and }; } and . i. Add }, }, and 4; 3, }, and }; }, 4, and }; 1b, å, and }. j. If you should spend ļ of your time in school, 4 in practising music, and f in sewing and studying, what time would you spend in all ? k. Owning of a paper-mill, I bought the shares of two other persons who owned to and frespectively. What part of the mill did I then own? 219. From the above examples may be derived the following Rule. To change fractions to equivalent fractions having the least common denominator : 1. For the common denominator, find the least common multiple of the given denominators. 2. For the new numerators, multiply the numerator of each fraction by the number by which you multiply its denominator to produce the common denominator. NOTE. If the number to multiply the numerator by is not readily seen, it may be found by dividing the common denominator by the denominator of the given fraction. 220. From what we have now learned of the addition of fractions, we may derive the following Rule. To add fractions : 1. If they have a common denominator, add their numerators. 2. If they have not a common denominator, change them to equivalent fractions that have a common denominator, and then add their numerators. 221. Examples for the Slate. (30.) 4 + 2x + 1 + f = ? (35.) 1 + 15 + }} = ? Add the integers and fractions of the following, and similar examples, separately : 40. In my furnace there were burned 24 tons of coal in December, 2 tons in January, and 3 in February. How many tons were burned in all ? (41.) 724 +164 +184 +234 +3775 = ? 42. A horse travelled 4314 miles in one day, 524 the next, 3684 the third, and 4037 the fourth. How far did he travel in all ? 43. A merchant had three barrels of sugar, the first containing 2474 pounds; the second, 2294 pounds; and the third, 2607 pounds. What was the weight of the whole ? For other examples in addition of fractions, see page 123. * What operation should first be performed on this fraction ? |