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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 i. 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 writteh thus, 3; 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 - - - 2

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

Illustrate by a line, divided into 7 equal parts, and three of the parts taken. In the same way illustrate the meaning of the fractions §, $, or. .

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 or, 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.; * or by blackboard, 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 writ. ing fractions Having analyzed the question, the pupil may write the required fraction on the blackboard.

See Section VIII. A.

In 2 how many times #7° how many times of how many times of how many times of how many times + 2 how many times of 8 how many times #2 In 2%f how many times of . In 34 how many times #2 10. Reduce 4} to an improper fraction. † 11. Reduce 3} 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. See Key.

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13. Reduce 63 to an improper fraction. 14. Reduce 8; to an improper fraction. 15. Iteduce 9} to an improper fraction.

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. 12 is # of what number 1 15 is ; of what number 1 18 is # of what number 1 20 is # of what number 7 24 is ; of what number 1 28 is ; of what number 1 . 30 is or of what number? 3 is # of what number 1 4 is ; of what number 7 5 is 3 of what number 1 8 is ; of what number 1 9 is # of what number 1 . 17 is ; of what number 1 . 25 is 3 of what number 1 . 38 is ; of what number 1 19. 43 is ; of what number 1 20. 54 is o 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 1 3. A man sold # of a gallon of wine for 40 cents; what was that a gallon 1 4. If it will take 14 yards of cloth to make a coat, how many yards will it take to make 7 coats 1 5. If I horse consume 3} bushels of oats in 2 days, how much would 2 horses in 5 days 1 6. If when the days are 94 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 73 dollars a yard, and received 8 firkins of butter at 63 dollars a firkin; how much was then due to him 1 8. Two men are 38 miles apart, and are travel

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