14 If 3-eighths of a bale of cotton be worth 17 dol lars, what is the whole bale worth? 15. A laborer spent 30 cents a day, and still saved 3-sevenths of his wages. How much was his wages? 16. Suppose that I have read 5-ninths of the pages in a certain book, and there are 35 pages more to be read; -how many pages are there in the book? NOTATION OF FRACTIONS. Learners will now attend to the meaning of the words, FRACTION, DENOMINATOR, and NUMERATOR. A Fraction is any part of one. For example, one-half of an orange is a fraction of 1 orange; three-fourths of an orange is another fraction of 1 orange. In this book, fractions have been expressed by a number joined with a word; thus, 4-ninths. Fractions are commonly expressed by two numbers, standing one above the other, with a line between them; thus,, 17. What fraction is expressed, when there is a 4 with a 1 over it? 7 with 2 over it? 8 with 5 over it? 10 with 6 over it? 18. Which is the greater fraction; or ?or? or ?or? / or /? 19. Which is the greater fraction; or ? or ? oror? for? The Denominator of a fraction, is the number of equal parts into which a whole one is divided. For example, if a whole orange be divided into 4 equal parts, the denominator is 4; the parts being denominated fourths. The Numerator of a fraction is the number which shows how many of the equal parts the fraction expresses. For example, the fraction expresses 3 of the four equal parts; therefore 3 is the numerator. 20. What numerator, and what denominator, would express the fraction, four-fifths? two-eighths? ninths? one fifteenth? five-eighteenths? six When the numerator is equal to the denominator, thus,, then the fraction is equal to 1; as 4-fourths of an orange, when joined together make 1 orange. When the numerator is greater than the denominator, thus,, then the fraction is equal to as many times 1 as the denominator is contained times in the numerator. 21. How many times 1, [how many whole ones], in ? in 2? in 16? in 20? in 28? in 2? 22. How many times 1, and what fraction over, in 1⁄2? in in? in ? in 36? in 15? 23. Where have you observed the numerator of a fraction to stand; above, or below the denominator ? SECTION 25. 1. James has of a dollar, and Henry has of a dollar:- which of them has the most money? Compare the fraction with other fractions. 2. is equal to how ma ny fourths? 3. is equal to how many sixths? 4. is equal to how many eighths? 5. is equal to how many tenths? 6. is equal to how many twelfths? how many sixteenths? how many twentieths? 7. Edward broke a slate pencil into 3 equal pieces, and Albert broke one into 6 equal pieces. How many of Albert's pieces were equal to 1 of Edward's pieces? Compare the fraction with other fractions. 8. is equal to how many sixths? 9, is equal to how ma ny ninths? 10. is equal to how many twelfths? how many eighteenths? how many thirtieths? Suggestion. of 1 is equal to of 12-twelfths. 11. is equal to how many eighths? how many twelfths? how many sixteenths? 12. is equal to how many tenths? how many twentieths? how many twenty-fifths? 13. is equal to how many twelfths? how many eighteenths? how many thirtieths? 14. is equal to how many fourteenths? how many twenty-eighths? how many thirty-fifths? 15. is equal to how many twentieths? Solution. is equal to, is equal to 3 times 5twentieths, which is 15. 16. is equal to how many twelfths? 17. A boy, who had of an orange, cut each fifth into 2 parts, (making tenths); his brother gave him more. What fraction of an orange had he at last? 18. Into how many parts must you cut a sixth of an orange, to make the parts eighteenths?...... Why? 19. is equal to how many eighteenths? 20. Change to fourteenths, and then add to it. 21. is equal to how many twenty-fourths? 22. Change to eighteenths, and then take from it. SECTION 26. 1. What is meant by a Fraction?. -How is a frac tion commonly expressed?.What is the Denominator of a fraction ?.- -What is the Numerator? 2. If the denominator of a fraction be 9, and the numerator 7, how should these numbers be written ?and what would the fraction be called in reading it? 3. Suppose two fractions have numerators alike, and denominators different-which is the greater fraction-that with the greater, or the smaller denominator? 4. Suppose two fractions have denominators alike, and numerators different-which is the greater fraction -that with the greater, or the smaller numerator? Observation. If an orange be cut into eighths, and then 4 of the eighths be joined together, these 4-eighths become 1-half of an orange. And thus the fraction,, when reduced to its lowest terms, is . 5. Reduce to its lowest terms-that is, find the lowest numerator and denominator, that will express a quantity equal to 4. 6. Reduce to its lowest terms. 7. Reduce & to its lowest terms. 8. Reduce to its lowest terms. 9. Reduce to its lowest terms, Reduce 4. Observation. A fraction is reduced, by dividing the numerator and denominator by any number, which will divide them both without a remainder. For example, we reduce thus; 2 in 6, 3 times, 3 is a new numerator: 2 in 10, 5 times, 5 is a new denominator. 8 10. Reduce each of the following fractions to its lowest terms. †. ů. 11⁄2. 12. 12. 12. 1. 11. Stephen's knife cost of a dollar, and John's cost of a dollar. Whose knife cost the most? 12. Reduce the fractions, and 10 to their lowest terms, and then add them together. 13. Reduce, and then add together, 16 and 12. 14. Reduce, and then add together, and . SECTION 27. 36 1. of a watermelon was divided equally among S boys. What part of the whole melon did 1 boy receive? 2. ofis equal to what part of 1? Solution. If be divided into 3 equal parts, it will take 12 such parts to make a whole one. ofis. [ times is 12.] 4 3. What part of a whole one is of ? Therefore, Illustration. If of an orange were cut into 6 equal parts, it would take 5 times 6 such parts to make a whole orange. Operation. times is 3 4. What part of 1 is of ? of? of? of of of ? 5. If 18 dollars [18 and dollars] be divided equally among 3 men, what will each man receive? 6. What is of 184? Sol. of 18 is 6; ofis 12; 6 plus 12 is 612. 7. What is of 30 of 242 of 18? 8. A boy, having of a dollar, paid of his money for a knife. What part of a dollar did he pay? 9. of is equal to what part of 1? Solution. One-fourth of is equal to ; three-fourths ofis equal 3 times, which is . 10. What part of 1 is of ? ? of? of? of? of? % of 11. A girl having of a dollar, paid of her money for a book. What part of a dollar did she pay? 12. ofis equal to what part of 1? T Solution. of one-fourth [ times is] is equal to ofis 3 times, which is . 13. Which is the greater fraction of a dollar,-3 of of a dollar, or, of of a dollar? - 14. What part of 1 is of ? of? of? 18 of 2 of 5 of 7? 15. If 4 cloaks are to be made from 123 yards of cloth, how many yards must be put into each cloak? 16. What is of 12? of his money 18. A boy having of a dollar, paid for a book. What part of a dollar did he pay? 19.of is equal to what part of 1? Solution. 1s 2 times, or is equal to ofis ; of . |