To change Improper Fractions to Integers or to Mixed Numbers. 204. A fractional number, the numerator of which equals or exceeds the denominator, is called an improper fraction. 205. ILLUSTRATIVE EXAMPLE III. Change and as far as possible to integers. WRITTEN WORK. (1.) 12) 60 Ans. 5 (2.) 12) 47 Ans. 311 Explanation.-(1.) Since 12 twelfths make a unit, in 60 twelfths there are as many units as there are 12's in 60, which is 5. Ans. 5. (2.) In there are as many units as there are 12's in 47, which is 3 and 1. Ans. 311. 206. The number 311 consists of an integer and a fraction. A number con sisting of an integer and a fraction is a mixed number. 207. Oral Exercises. a. Change to integral numbers: ; ; ; 25; 4; &; 14; 21; 28; 36; 42; 56; 72; ††; &; f2; 45; §8. b. Change to mixed numbers: ; ; ; 15; 12; 22; 28; 35; 43; 1; 5; 4; 17; f8; 93; 58; 71. c. Change to integers or to mixed numbers: ; ; ; 18; 55; Ab; 54; ff; up; 188; 92; sp. 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. To change an Integer or a Mixed Number to an Improper Fraction. 210. ILLUSTRATIVE EXAMPLE IV. Change 231 to fourths. WRITTEN WORK. 231 = 23 Ans. 4 93 Explanation. Since in 1 there are 4 fourths, in 23 there are 23 times 4 fourths, or 92 fourths, which, with 1 fourth added, are 93 fourths. Ans. 23. 211. Oral Exercises. a. Change to improper fractions: 2; 3; 23; 5; 22; 31; 6; 5; 5; 7; 7; 83; 85; 91; 98; 10. b. Change to improper fractions: 2; 28; 3; 3; 4; 48; 5; 93; 62; 7; 83; 94; 44; 43; 8; 71. 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 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.) 694. (24.) 7641. (27.) Change 48 to ninths. (28.) Change 567 to tenths. (23.) 109. (26.) 663. For other examples in reduction of fractions, see page 123. (29.) Change 93 to forty-thirds. ADDITION OF FRACTIONS. To add Fractions having a Common Denominator. 214. ILLUSTRATIVE EXAMPLE I. Add of an apple, of an apple, and of an apple. Ans. § 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 How do you add fractions which have a common denominator? To add Fractions not having a Common Denominator. 217. ILLUSTRATIVE EXAMPLE. Add 1⁄2, 1, and 7. WRITTEN WORK. 2 × 3 × 3 × 5 = 90 1. c. denom. 5 15 8 = 8 x18=38 # = 3×18=48 X 6 78 = 78x @ = 38 15× Ans. 157187. Explanation. To be added, these fractions must be changed to like fractions, or to fractions having a common denominator. (Art. 215.) The new denominator must be some multiple of the given denominators. A convenient multiple is their least common multiple, which is 90. (Art. 182.) denominator 6 must be multiplied by 3x5, or 15; hence the numerator 5 must be multiplied by 15. (Art. To change to 90ths, the 199.) Thus, is found to equal 8. In a similar way will be found to equal 8, and to equal 3. Adding these fractions, we have 157, or 187, for the sum. NOTE. When the denominators are prime to each other, the new denominator will be the product of all the denominators, and the new numerators will be found by multiplying each numerator by the product of all the denominators except its own. g. Add and ; } and 1; } and †; § and ; and ; and ; and ; and §. h. Add and ; and ; and ; and }; } and . i. Add,, and 4; 3, 1, and ; 1, §, and ; 1, 2, and 1. j. If you should spend of your time in school, in practising music, and 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 and respectively. 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 To add fractions: Rule. 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. 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 33 in February. How many tons were burned in all ? (41.) 72+16 +18+231 +377 = ? 42. A horse travelled 43 miles in one day, 524 the next, 364 the third, and 4037 the fourth. How far did he travel in all ? 43. A merchant had three barrels of sugar, the first containing 247 pounds; the second, 229 pounds; and the third, 260 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? |