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Q. What is the 5 (in §) called?

A. Numerator.

Q. Why so called?

A. Because it numerates or numbers the parts.

Q. Which is the numerator, then?

A. The number above the line.

Q. Which is the denominator?

A. The number below the line.

Q. What, then, does the denominator show?

A. The number of parts a unit, or any thing, is divided into.

Q. What does the numerator show?

A. How many parts are taken, or used.

Q. In the expressions, 16, 12, o, which are the numerators, and which are the denominators?

Q. If you own of a vessel, how many parts is the vessel sup posed to be divided into? and how many parts do you own? A. 40 parts, and I own 28 parts.

Q. Is of an apple more than of it?

Q. What fraction, then, is greater than ? Than? Than{? Than? Than? What fraction is less than ? Than? Than ? Than?

Q. From these remarks, what appears to be a correct definition of Tractions?

A. They are broken parts of a whole number.
Q. How are they represented?

A. By one number placed above another, with a line drawn between them.

in Simple Division, you recollect, that the remainder was represented in like manner; what, then, may justly be considered the origin of fractions?

A. Division.

Q. What may the numerator be considered?

A. The dividend.

Q. What may the denominator be considered?

A. The divisor.

Q. What, then, is the value of a fraction?

A. The quotient of the numerator divided by

the denominator.

Q. What is the quotient of 1 dollar divided among 2 men ?
A..

Q. What is the quotient of 7 divided by 8?

4. 1.

Q. How, then, are fractions represented?

A. By the sign of division.
Q. What does express?

A. The quotient, of which 2 is the dividend.

3 is the divisor.

1. If 3 apples be divided equally among 8 boys, what part of one apple will each boy receive? 1 apple among 8 boys would be of an apple apiece, and 3 apples would be 3 times as much; that is, of an apple apiece. Ans. §.

8

2. If 4 oranges be divided equally among 8 boys, what part of an orange is each boy's part? 1 orange among boys, and 4 oranges are 4 times as much; that is, , Ans. If 2 oranges among 7 boys? A. . 9 oranges among 13 boys? 20 oranges among 37 boys?

3. One orange among 2 boys is of an orange apiece, how much is 1 divided by 2, then? Ans. . How much is 1 divided by 3? A. §. The quotient of 5 divided by 6? A.. Of 3 by 5? Of 7 by 9? Of 8 by 13? Of 11 by 15?

4. What part of one apple is a third part of 2 apples? A third part of one apple is, and a third part of 2 apples must be twice as much; that is, of 1 apple. A..

5. What part of 1 apple is one fourth (4) part of 3 apples? of 3 apples is 3 times as much as of 1 apple; that is, of 1 apple. A. §.

6. What part of 1 apple is of 3 apples? A. §. What 'part of 1 apple is of 4 apples? A. . of 4 apples is what part of 1 apple? Ans. .

A PROPER FRACTION. Q. We have seen that the denominator shows how many parts it takes to make a whole or unit; when, then, the numerator is less than the denominator, is the fraction greater, or less, than a whole thing or unit?

A. It must be less.

Q. What is such a fraction called?

A. A Proper Fraction.

Q. How may it always be known?

A. The numerator is less than the denominator.
Q. What kind of fractions are,,, &c.?

AN IMPROPER FRACTION. Q. When the numerator is as large, or larger than the denominator, as, &,,, it is plain, that the fraction expresses 1 whole, or more than 1 whole: what is such a fraction called?

A. An Improper Fraction.

Q. How may it be known?

A. The numerator is greater than the denomi

nator.

Q. What kind of fractions are, 42, §, &c. ?

A MIXED NUMBER. Q. What is a mixed number?

A. A fraction joined with a whole number.

Q. What kind of fractions are 152, 167, &c. ?

Q. What kind of fractions are each of the following expressions, viz. 158, §, 21, 8, 18, 73, 50?

¶ XXXV. TO CHANGE AN IMPROPER FRACTION TO A WHOLE OR MIXED NUMBER.

1. How many whole apples are there in 6 thirds (§) of an apple? In 8 quarters ()? In 2? In 16? In 24? In 28? In 488?

2. How many weeks in of a week? In 28? In 42? In 56? In 84 ?

3. How many pints in gills? In 32 gills? In 48 gills? In 120 gills?

4. How much is § of a dollar? A. $1. Is g? A. 1 and 1. Is ? Is 16? Is 7? Is 24? Is 25?

Q. What is the finding how many whole things are contained in an improper fraction called?

A. Reducing an improper fraction to a whole or mixed number.

1. James, by saving

of a dollar a day, would save in 33

days; how many dollars would that be?

OPERATION.

16) 33

Ans. 2 dollars.

In this example, as 18 make 1 dol lar, it is plain, that as many times as 16 is contained in 33, so many dollars it is; 16 is contained 2 times and 1 over; that is, 2 dollars.

RULE. Q. What, then, is the rule for reducing an improper frac tion to a whole or mixed number?

A. Divide the numerator by the denominator.

More Exercises for the Slate.

2. A regiment of soldiers, consuming of a barrel of pork a day, would consume in 28 days 28 of a barrel; how many barrels would that be? A. 5 barrels.

3. A man, saving of a dollar a day, would save in 365 days 26; how many dollars would that be? 4. Reduce 1881 to a mixed number.

to a mixed number.

A. $73.

A. 20.

A. 721.

5. Reduce

6. Reduce

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10. Reduce to a whole number. A. 144.

1 XXXVI. TO REDUCE A WHOLE OR MIXED NUMBER TO AN IMPROPER FRACTION.

1. How many halves will 2 whole apples make? Will 3? Will 4 Will 6? Will 20? Will 100?

2. How many thirds in 2 whole oranges? In 2? In 23 ? In 3? In 3? In 8? In 12?

3. A father, dividing one whole apple among his children, gave them of an apple apiece; how many children were there?

4. James, by saving of a dollar a day, found, after several days, that he had saved 13 of a dollar; how many 8ths did he save? and how many days was he in saving them?

5. How many 7ths in 2 whole oranges? In 24? In 24? In 34?

This rule, it will be perceived, is exactly the reverse of the last, and proves the operations of it.

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RULE. Q. What, then, is the rule for redueing a mixed or whole number to an improper fraction?

A. Multiply the whole number by the denominator of the fraction.

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Q. What do you add to the product?

A. The numerator.

Q. What is to be written under this result?

A. The denominator.

More Exercises for the Slate.

2. What improper fraction is equal to 20?
3. What improper fraction is equal to 7212?
4. What improper fraction is equal to 4?
5. What improper fraction is equal to 123?
6. What improper fraction is equal to 16?

A. 1201

A. 874
Ꭿ. 38 .

A. 38.

A. 197. 7. What improper fraction is equal to 17? A. 189. 8. What improper fraction is equal to 144? A. 1729.

9. Reduce 30 pounds to 20ths. As of a pound1s, 2s., the question is the same as if it had been stated thus ·

30 5 s. how many shillings? A. 905-605 shillings.

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weeks, how many 7ths? A. 191101 days.

pecks, how many 8this?

A. 21=211 quarts.

¶XXXVII. TO REDUCE A FRACTION TO ITS LOWEST

TERMS.

Q. When an apple is divided into 4 parts, 2 parts, or, are evidently of the apple: now, if we take, and multiply the 1 and 2 both by 2, we shall have again; why does not this multiplying alter the value?

A. Because, when the apple is divided into 4 parts, or quarters, it takes 2 times as many parts, or quarters, to make one whole apple, as it will take parts, when the apple is divided into only 2 parts, or halves: hence, multiplying only increases the number of parts of a whole, without altering the value of the fraction.

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