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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 manner that we may always know what each of them is intended to express.

One half is usually written thus, }; 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 pumber of parts into which one is divided. The number

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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 to, 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 allowed to proceed any

farther. Particular care must be taken to make him understand what the denominator signities, 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 ]? Ans.
2. In 3 how many times 4? Ans. f.
3. In 2 how many times ? Ans. .
4. In 4 how many times ?
5. In 6 how many times † ?
6. In 7 how many times ?
7. In 8 how many times ?
8. In 21t how many times ?
9. In 37 how many

times ?
10. Reduce 4to an improper fraction. I
11. Reduce 34 to an improper fraction.
12. Reduce 5to an improper fraction.

* When the numerator is larger than the denominator, the fraction is called an improper fraction. + 24 is read two and 1 half. It is called a mixed ramber.

That is, to find how many fifths there are in four and 1 fiftha

13. Reduce 69 to an improper fraction. 14. Reduce 83 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. \ are how many times 1 ? 7. are how many times 1 ? 8. 43 are how many times 1? 9. z are how many times 1 ? 10. f7 are how many times 1 ?

See Section IX.

A. 1. How much is 3 times 1 ? 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 f? 10. How much is 9 times ?

B. 1. How much is 3 times 21?
2. How much is 4 times 33 ?
3. How much is 5 times 64?
4. How much is 6 times 45?
5. How much is 7 times 54 ?
6. How much is 8 times 6*?
7. How much is 4 times 10;?

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4. 12 is of what number? 5. 15 is of what number? 6. 18 is & of what number? 7. 20 is & of what number? 8. 24 is of what number? 9. 28 is of what number? 10. 30 is Bo of what number? 11. 3 is of what number? 12. 4 is of what number? 13. 5 is of what number? 14.

8 is of what number ? 15. 9 is of what number? 16. 17 is ģ of what number? 17. 25 is of what number? 18. 38 is of what number? 19. 43 is of what number? 20. 54 is y of what number?

Miscellaneous Examples. 1. A man sold 8 yards of cloth for 34 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 17 yards of cloth to make a coat, how many yards will it take to make 7 coats ?

5. If i horse consume 37 bushels of oats in 2 days, how much would 2 lorses in 5 days ?

6.' If, when the days are 91 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 7 dollars a yard, and received firkins of butter at 60 dollars, å firkin; how much was then due to him?

8. Two men are 38 miles apart, and are travel

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