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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 shows the number of parts used. Thus io, 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 timesį? Ans. 4.*
2. In 3 how many times į? Ans. s.
3. In 2 how many times 1 ? Ans.
4. In 4 how many times ž?
5. In 6 how many times . ?
6. In 7 how many times ; ?
7. In 8 how many times } ?
8. In 2+ how many times į?
9. In 31 how many times ?
10. Reduce 4} to an improper fraction. I
11. Reduce 34 to an improper fraction.
12. Reduce 5% to an improper fraction.

* When the numerator is larger than the denominator, the fraction is called an improper fraction. 12 1-2 is reid two and 1 half. It is called a mixed number. That is, to find how many fifths there are in four and 1 fifth.

13. Reduce 6% to an improper fraction. 14. Reduce 8% to an improper fraction. 15. Reduce 94 to an improper fraction.

B. 1. are how many times ? 2.

are how many times 1 ? 3. į are how many times 1 ? 4. are how many times 1 ? 5. į are how many times 1 ? 6. Z are how many times 1 ? 7. * are how many times 1 ? 8. 3 are how many times 1 ? 9. 6 are how many times 1 ? 10. 13 are how many times 1 ?

See Section IX.

A. 1. How much is 3 times 1 ? 2. How much is 4 times į? 3. How much is 3 times ? 4. How much is 4 times ? 5. How much is 5 times ? 6. How much is 6 tinies 4 ? 7. How much is 8 times in ? 8. How much is 9 times ? 9. How much is 10 times ? 10. How much is 9 times ?

B. 1. How much is 3 times 21 ?
2. How much is 4 times 3?
3. How much is 5 times 64 ?
4. How much is 6 times 4?
5. How much is 7 times 54 ?
6. How much is 8 times 69?
7. How much is 4 times 104?

[blocks in formation]

B. 1. 4 is of what number?
2. 6 is i of what number?
3. 8 is of what number?

4. 12 is of what number? 5. 15 is of what number? 6. 18 is of what number ? 7. 20 is 5 of what number? 8. 24 is of what number? 9. 28 is į of what number? 10. 30 is of what number? ll. 3 is of what number? 12. 4 is z of what number? 13. 5 is of what number? 14. 3 is of what number ! 15. 9 is of what number? 16. 17 is of what number? 17. 25 is of what number? 18. 38 is of what number ? 19. 43 is of what number? 20. 54 is of what number ?

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 sold of a gallon of wine for 40 cents; what was that a gallon ?

4. If it will take l] yards of cloth to make a coat, how many yards will it take to make 7 coats ?

5. If I horse consume 31 bushels of oats in 2 days, how much would 2 horses in 5 days ?

6. If when the days are 9.4 hours long a man pero form a journey in 10 days, in how many days would he perform it when the days are 12 hours long?

7. A man soid 8 yards of cloth for 7 dollars a yard, aud received 8 firkin's of butter at 6 dollars a firkin; how much was then due to him ?

8. Two men are 33 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 ? How much in 2 hours ? In how many hours will they meet? 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 įto another, how much more do you give the first, than the second ?

2. ¿ of an orange is how many of an orange ?

3. If you give of an orange to one boy, and I to another, how many would you give away ? How many i would you have left?

4. { and are how many ? 5. A man gave to one labourer of a bushel of wheat, and to another; how many of a bushel did he give to both ? How many bushels ?

6. j and I are how many 1 ? How many times 1 ?

7. A man gave į of a barrel of flour to one man, and of a barrel to another; to which did he give the most?

8. } is how many ?

9. A man bought of a bushel of wheat at one time, and of a bushel at another; at w! ich time did he buy the most ?

10. Į is how many $?

11. A man bought į of a yard of cloth at one time, and of a yard at another; at which time did he buy the most ?

12. 1 are how many $ ? 13. A man wished to give ; of a bushel of wheat

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