horse for 73 dollars; what was each man's share of the gain? 142. A farmer put 85 pounds of butter into 11 boxes; how many pounds did he put into each box? 143. If a man can do a piece of work in 98 hours, how many days of 12 hours each, will it require for him to do it? CHAPTER VI. § 20.-We have already learned, (§ 19,) that parts of a unit (§ 18) are called fractions. Fractions, as well as whole numbers, may be represented by figures. To do this, two numbers are required; one to show the number or size of the parts into which the unit is broken, and the other to show how many of those parts are taken. These numbers are usually written one above the other with a line between them. The number below the line shows into how many parts the unit is divided or the size of the parts, and is called the DENOMINATOR. The number above the line shows how many parts are taken, and is called the NUME RATOR. ONE HALF is represented by the character . One third is written ; two thirds ; three thirds 3, which equals 1. 1 2 One fourth is written ; two fourths ; three fourths 3 4 One fifth is written; two fifths; three fifths, &c., &c. One sixth is written ; two sixths &, &c., &c. 1 6 Illustration.— of an apple denotes that the apple is cut into 7 pieces, and that 5 of those parts are taken. Note.-The teacher should illustrate by actually cutting an apple or some other convenient object into seven equal parts. § 21.-It will be observed that the fraction takes its name from the denominator; (§ 20;) thus when the unit is divided into eight equal parts, they are called eighths, and one part is represented; when the unit is divided into nine equal parts, the parts are called ninths; and one part is represented, &c., &c. Note.-Care must be taken to make the pupils familiar with the mode of expressing fractions by figures. Be sure they understand fully what is denoted by the denominator, which should be first explained; also by the numerator. The pupil should be required to write the fractions on the black-board, and made frequently to explain the meaning of each term. § 22.-If the numerator is larger than the denominator, the value of the fraction is greater than a unit, and the fraction is called IMProper. If the numerator is less than the denominator, the value of the fraction is less than a unit, and the fraction is called proper. REVIEW.-What are fractions? How are they expressed by figures? What is the number below the line called and what does it denote ?--the number above the line? From which of these numbers does the fraction take its name? Why? What is an improper fraction? What is a proper fraction Will you write the fraction one ninth on the black-board? One seventh? One fourth? Five sevenths? Eight tenths? Nine fifths? Seven elevenths? Five thirds, &c., &c. Is nine 3 tenths a proper, or an improper fraction? Why? Is a proper or an improper fraction?? 11 ? 11 ? &c. 4 9 12 1. If of an orange cost 3 cents, how much will a whole orange cost? SOLUTION.-If of an orange cost 3 cents, two, the number of halves in 1, will cost 2 times 3 cents; 2 times 3 cents are 6 cents. Therefore, one orange would cost 6 cents. 2. A man sold of a cord of wood for 4 shillings; how much would a whole cord cost? In one cord of wood, how many thirds of a cord? how many fourths? how many sevenths? ninths? halves? 3. At 4 dollars a barrel, what will 3 and 4 barrels of flour cost? 3 times 4 and 4 of 4 are how many? 4. If a laborer can earn 5 dollars a week, how much can he earn in 4 weeks and of a week? 4 times 5 and of 5 are how many? 5. 3 times 6 and 6. 4 times 7 and 7. 5 times 8 and 8. 6 times 9 and 9. 7 times 10 and of 6 are how 10 10. At 12 cents a pound, what would 8 pounds and a pound of sugar cost? are how many? 11. 9 times 11 and 3 of 11 are how II 12 12. 10 times 12 and 2 of 12 are how many? are how many? 13. 4 times 9 and 3 of 9 are 14. 7 times 8 and 3 of 8 are how many ? 15. 8 times 5 and of 5 are how many? 16. A man bought 3 barrels of cider at 4 dollars a barrel, and to pay for it he gave flour at 3 dollars a barrel; how many barrels did it take? 3 times 4 is how many times 3? 17. A lady bought 4 yards of cloth, at 21⁄2 dollars a yard, and paid for it with thread at 2 dollars a pound; how many pounds of thread did it take? 4 times 21 is how many times 2? 18. 5 times 6 and 7? 3 ? 9 ? of 6 are how many times 4? 19. 7 times 7 and 2 of 7 are how many times 5? 6? 8 9 10 11 ? 12? 10 20. 9 times 10 and 3 of 10 are how many times 12? 7? 8? 11? 12 21. 3 times 12 and of 12 are how many times 7? 6? 9? 10? 5? 4? 22. 8 times 5 and of 5 are how many times 6 ? 9? 3 ? 8? 7? 5? SOLUTION.-8 times 5 is 40. of 5 is 1; is 4 44 is as many times 6 as the number of times that 6 is contained in 44; 6 in 44, 72 times. Therefore, 8 times 5 and of 5 are 72 times 6. times 1; 4 times 1 is 4; 40 and 4 are 44. 23. 9 times 6 and 4? 8? of six are how many times 7? 24. 3 times 4 and of 4 are how many times 2 ? 5? 6 ? 7? 25. 10 times 5 and 8? 9 ? 11 ? 12? 26. 8 times 9 and 6? 7? 8? 10? 27. 5 times 10 and 7 ? 6 ? 8 ? 7? 12? of 5 are how many times 7? of 9 are how many times 5? 5 of 10 are how many times 10 28. 11 times 11 and of 11 are how many times 7? 8? 6? 9? 10? 12 29. 12 times 12 and of 12 are how many times 6 ? 7 ? 8? 9? 11? 5 30. 5 times 12 and 12 of 12 are how many times 7? 6 ? 9 ? 8? 11? 31. If 3 yards of cloth cost 9 dollars, what will 4 yards cost? 32. If 3 fourths of a barrel of flour cost 6 dollars, what is that a barrel? 33. If I pay 12 dollars for 6 sheep, what must I pay for 7, at the same rate? 8? 9? 12? 11? 34. If 3 men can do a piece of work in 5 days, in what length of time can 7 men do it? men? 4 men ? 10 35. If 27 dollars are paid for 9 barrels of cider, how many yards of cloth, at 2 dollars a yard, must be given for 7 barrels of cider? § 23.-1. How many fourths are there in 1? 2? 31 52 72 91? ? 2. How many fifths in 1 in 2? 33? 4? 54? 7름? 8!? 10층? 3. How many sixths in 3? 2? 55? 74? 101? 88 ? 77 ? 4. How many sevenths in 3? 4? 37? 83? 115? 12? 95? 5. How many eighths in 2? 51 732 87 91 ? 7급? 3급? § 24.-1. How many whole ones in in?? 12 ? 25 ? 2. 3 How many whole ones in 3? g? 11 ? 27 ? 31 ? 47 ? 50 ? 3 3. 6 How many whole ones in ? 21 ? 37 ? 42 ? 54 ? 6 4. 2 6 6 6 How many whole ones in? 20? 40? 57 ? 37? 97 ? 86 ? 9 9 9 5. How many whole ones in 10? 13? 55 ? 962 99 ? 10 10 10 84 10 |