under the line is called the denominator, because it gives name to the fraction; and the number above the line is called the numerator, because it sho^ viie number of parts used. Thus T\, 10 is the denominator and 3 the numerator. N. B. The pupil must be made familiar with this mode of expressing fractions, and must be able to apply it to any familiar objects; as apples, oranges, &c.; and to the table, before he is allowed to proceed any farther. Particular care must be taken to make him understand what the denominator signifies, and what the numerator, as explained above. The denominator should always be explained first. The following examples are a recapitulation of some of the foregoing sections, for the purpose of showing the application of the above method of writing fractions. See Section VIII. A. A. 1. In 2 how many times 11 Ans. f * 2. In 3 how many times | 1 Ans. $. 3. In 2 how many times £? Ans. f. 4. In 4 how many times ^ 1 5. In 6 how many times \ 1 6. In 7 how many times } 1 7. In 8 how many times \ 1 8. In 2£t how many times \ 1 9. In 3£ how many times \ 1 10. Reduce 4i to an improper fraction-! 11. Reduce 3f to an improper fraction. 12. Reduce 5 J to an improper fraction. * When the numerator is larger than the denominator, the fraction i called an improper* fraction. 12 1-2 is read two and 1 half. It is called a mixed number. hat is, to find how many fifths there are in four and 1 fifth. 13. Reduce 6J to an improper fraction. 14. Reduce 8T\ to an improper fraction. 15. Reduce 9f to an improper fraction. B. 1. |- are how many times 1? 2. | are how many times 1? 3. \ are how many times 11 4. | are how many times 1? 5. f are how many times 1? 6. y are how many times I?' 7. y are now many times 1? 8. y are how many times 1? 9. y are how many times 1? 10. \£ are how many times 1? jSfee Section IX. A. 1. How much is 3 times £ 1 2. How much is 4 times i? 3. How much is 3 times f 1 4. How much is 4 times ^? 5. How much is 5 times f 1 6. How much is 6 times J? 7. How much is 8 times T3y? 8. How much is 9 times % 1 9. How much is 10 times f? 10. How much is 9 times f 1 B. 1. How much is 3 times 2J1 2. How much is 4 times 3|? 3. How much is 5 times 6|? 4. How much is 6 times 4|? 5. How much is 7 times 5f? G. How much is S times 6f t 7. How much is 4 times 10^ 1 8. How much is 9 times 7|? 9. How much is 8 times 9$? 10. How much is 10 times 7f 1 Sfec Sections V. & X. 1. What is £ of 6 1 3. What is J of 8? 5. What is | of 9? 7. What is \ of 14 2 9. What is f of 5? 11. What is | of 7? 13. What is \ of 17? 15. What is £ of 27? 17. What is $ of 47? 19. What is 4 of 65? 2. What is £ of 6? 4. What is | of 9? 6. Whafisjof 101 8. What is | of 5? 10. What is £ of 7? 12. What is f of 35? 14. What is f of 26? 16. What is £ of 37? 18. What is | of 42? 20. What is \ of 75 ) See Sections VI. & XI. A. 1. 2 is i of what number? 2. 4 is £ of what number? 3. 8 is | of what number? 4. 1^ is - of what number? 5. 2 J is £ of what number? 6. 4f is i of what number? 7. 6f is ^ of what number? 8. 7f is i of what number? 9. 8J is | of what number? 10. 9T\ is TV of what number? B. 1. 4 is § of what number? 2. 6 is f of what number? 3. 8 is of what number? 4. 12 is % of what number? 5. 15 is f of what number 1 6. 18 is f of what number? 7. 20 is | of what number 1 8. 24 is | of what number? 9. 28 is | of what number? 10. 30 is T57 of what number 1 11. .3 is f of what number? 12. 4 is f of what number? 13. 5 is 4 of wjiat number? 14. 8 is | of what number? 15. 9 is f of what number? 16. 17 is | of what number? 17. 25 is £ of what number 1 18. 38 is * of what number? 19. 43 is f what number? 20. 54 is V° of what number 1 Miscellaneous Examples. 1. A man sold 8 yards of cloth for 3£ dollars a yard; what did it come to? 2. A man sold a horse for 76 dollars, which was | of what it cost him; how much did it cost him? 3. A man spld J of a gallon of wine for 40 cents; what was that a gallon? 4. If it will take If yards of cloth to make a coat, how many yards will it take to make 7 coats? 5. If 1 horse consume 3i bushels of oats in 2* days, how much would 2 horses in 5 days? 6. If when the days are 9£ hours long a man perform a journey in 10 days, in how many days would he perform it when the days are 12 hours long? 7. A man sold 8 yards of cloth for 7j dollars a yard, and received 8 firkins of butter at 6| dollars a firkin; how much was then due to him 1 8. Two men are 38 miles apart, and are travel ing towards each other, one at the rate of 3 miles an hour, the other 2 miles; how much do they approach each other in an hour 1 How much in 2 hours 1 In how many hours will they meet 1 At what distance from each place from which they set out? SECTION XIII. A. 1. If you give \ of an orange to one hoy, and \ to another, how much more do you give the first, than the second 1 2. £ of an orange is how many £ of an orange 1 3. If you give \ of an orange to one boy, and % to another, how many \ would you give away t How many A would you have left 1 4. £ and £ are how many {1 5. A man gave to one labourer £ of a bushel of wheat, and f to another; how many | of a bushel did he give to both 1 How many bushels 1 6. ^ and J are how many £1 How many times 11 7. A man gave \ of a barrel of flour to one man, and l of a barrel to another; to which did he give the most 1 8. \ is how many \ 1 9. A man bought \ of a bushel of wheat at one time, and f of a bushel at another; at which time did he buy the most 1 10. i Is how many £ 1 11. A man bought f of a yard of cloth at one time, and f of a yard at another; at which time did he buy the most 1 12. | are how many } 1 13. A man wished to give J of a bushel of whea*. |