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67. 61 is 10 ninths of what number? 68. Bought 2 barrels and 1 fifth of a barrel of gin (that is, 11 fifths of a barrel) for 65 dollars; what was that a barrel?

69. 65 is 9 fifths of what number?
70. 71 is 8 elevenths of what number?

C. 1. A boy gave away 2 apples and i half, which was 1 fourth of all he had; how inany

had he ?

2. A man gave away 3 dollars, which was 2 fifths of all the money he had; how much had he?

3. A man sold a cow for fifteen dollars, which was 4 fifths of what she cost him; how much did he lose by the bargain ?

4. A man sold a piece of cloth for 37 dollars, which was 9 eighths of what it cost him ; how much did he gain by the bargain ?

5. There is a pole 3 fifths under water, and a feet out of the water; how long is the pole?

6. A man sold a piece of cloth for 47 dollars, by which bargain he lost 2 ninths of what the cloth cost him; how much did it cost him, and how much did he lose ?

Miscellaneous Examples. 1. If a staff 5 feet long cast a shadow 4 feet at 12 o'clock, what is the length of a pole that casts a shadow 67 feet at the same time?

2. If 53 gallon's of water, in 1 hour, run into a cistern containing 97 gallons, and 44 gallons run ou in an hour, in what time will it be filled ?

3. A man bought a cask of wine containing 75 gallons; 2 sevenths of it leaked out, and he sold the remainder for 1 dollar a gallon; how much did he poll it for?

4. A cask of wine cost 67 dollars; what is 5 eighths of it worth?

5. A man bought 9 oranges for 6 cents and 2 sevenths apiece, and sold them for 67 cents; what did he gain by the bargain?

6. A man bought 10 yards of broadcloth for 70 dollars; how must be sell it per yard in order to gain 14 dollars ?

7. If, when the days are 12 hours long, a man perform a journey in 3 days, how many hours is he in performing it?

8. If a man perform a journey in 36 hours, how many days would he be in performing it, when the days are 9 hours long?

9.. If, when the days are 11 hours long, a man can perform a journey in 5 days, in how many hours will he perform it? In how many days when the days are 9 hours long?

10. What number added to 2 fifths of 33 will make the number 177

11. How many yards of cloth, that is 1 quarter of a yard wide, will line 10 yards that is 3 quarters wide ?

12. 8 yards of cloth, that is 1 quarter wide, are equal to how many yards that is 4 quarters wide?

13. How many yards of cloth, that is 3 quarters wide, are equal to 7 yards that is 5 quarters wide ?

14. How many yards of cloth, that is 6 quarters wide, are equal to 37 that is 4 quarters wide ?

15. If a piece of cloth 5 quarters wide be worth $7 dollars, what is a piece of the same length, 3 quarters wide, worth?

16. If cloth 4 quarters wide is worth 8 dollars e yard, what is 1 yard of the same kind of cloth, that is 5 quarters uide, worth?

SECTION XII. Parts of one are called fractions. Fractions may be expressed by figures, as well as whole numbers. It requires two numbers to express a fraction; one to show into how many parts one is divided, and the other to show how many of those parts are used. For example, if we wish to express one half, (which means that one is divided into two equal parts, and that one part is used,) we must use the figure 2 to express that one is divided into two equal parts, and the figure 1 to show that one part is used. And these must be written in such a manaer that we may always know what each of them is intended to express.

One half is usually written thus, t; one number above a line, and the other below it. The number below the line shows into how many parts one is divided, and the number above the line shows how many parts are used.

One third is written
Two thirds
One fourth
Three fourths

Two fifths Example. of an apple signifies that the apple is to be cut into 7 equal parts, and that 3 parts are to be used.

Let us apply an example to Plate II. refers to a square divided into 8 parts, and signifies that 5 parts are to be used.

We may observe, that, when one is divided into 3. parts, the parts are called thirds; when one is divided into 4 parts, the parts are called fourths, &c. that is, the fraction takes its name from the number of parts into which one is divided. The number

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 3, 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.

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A. 1. In 2 how many times 4? Ans.
2. In 3 how many times 7? Ans. g.
3. In 2 how many times ? Ans. 4.
4. In 4 how many times ?
5. In 6 how many times ?
6. In 7 how

many

times ? 17. In 8 how many times ? 8. In 21t how many times 1 ? 9. In 37 how many times ? 10. Reduce 4} to an improper fraction. I 11. Reduce 34 to an improper fraction. 12. Reduce 5f to an improper fraction.

* When the numerator is larger than the denominator, the frastjan in oalled an improper fraction.

† 23 is road two and I half. It is called a mined member
$ That is, to find how many fifthis there are in four and 1 BA

13. Reduce 63.to an improper fraction. 14. Reduce 8.30 to an improper fraction. 15. Reduce 94 to an improper fraction.

B. 1. are how many times 1 ? 2. are how many times 1 : 3. Z are how many times 1 ? 4. f are how many times 1 ? 5. . are how many times 1 ? 6. 16 are how many times 1 ? 7. u are how many times 1 ? 8. 23 are how many times 1 ? 9. 26 are how many times 1 ? 10. 7 are how many times 1 ?

See Section IX.

A. 1. How much is 3 times ? 2. How much is 4 times ? 3. How much is 3 times ? 4. How much is 4 times ? 5. How much is 5 times ? 6. How much is 6 times ? 7. How much is 8 times it? 8. How much is 9 times ? 9. How much is 10 times 10. How much is 9 times g?

B. 1. How much is 3 times 21 ?
2. How much is 4 times 37?
3. How much is 5 times 64 ?
4. How much is 6 times 45 ?
5. How much is hy times 5%?
6. How much is 8 tiaes 69 ?
". How much is 4 times 104?

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