much is 1 divided by 2, then? Ans. . How much is 1 divid ed by 3? A. . The quotient of 5 divided by 6? 9. . 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. 4. 3.

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. 2.

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? 9. The numerator is less than the denominator.

Q. What kind of fractions are, 4, 7, &c.?

AN IMPROPER FRACTION. Q. When the numerator is us large, or larger than the denominator, as, §, 1†,‡, 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 denominator.

Q. What kind of fractions are §, 12, §, &c

A MIXED NUMBER. Q. What is a mixed number? A. A fraction joined with a whole number.

Q. What kind of fractions are 15, 167, &c.

n

What kind of fractions are each of the following expres

sions, viz. 15, 8, 22, 81, 18, 71, 50?

1 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? In 16? In 24? Ing? In 488 ?

2 How many weeks in of a week? in ? In 4? Iv 4&? In 94?

3. How many pints in gills? In 22 gills? In 8 gills

in 120 gills?

4. How much is of a dollar? A. $1. Is ? A. 1 and 1. Is? Is 16? Is 17? Is 24? Is 25?

Q. What is the finding how many whole things are contain ed 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 make 1 dollar, 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, 216 dollars.

RULE I. What, then, is the rule for reducing an improper fraction 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 2 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 365; how many dollars would that be? A. $73.

4. Reduce 181 to a mixed number. 9. 20. 5. Reduce, to a mixed number. A. 7212.

38

6. Reduce to a mixed number. A. 4.
7. Reduce 1 to a mixed number. A. 12.

8. Reduce 167 to a mixed number. A. 13

12 6284

9. Reduce to a mixed number. A. 2314.

10. Reduce 1728 to a whole number. A. 144.

↑ 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 27 In 2 Ia 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 2? In 24 ? In 34?

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

RULE I. What, then, is the rule for reducing a mixed or whole number to an improper fraction? A. Multiply the whole number by the denominator of the fraction.

II. What do you add to the product? A. The numerator. III. 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. 1881

60

A. 974.

A..

A. 36.

A. 197.

A. 189

- 1 s..

7. What improper fraction is equal to 17? 8. What improper fraction is equal to 144? A. 7. 9. Reduce 30 pounds to 20ths. As of a pound: 28., the question is the same as if it had been stated thus · in 30£ 5 s. how many shillings? A. 10. In 144 weeks, how many 7ths? 11. In 26 pecks, how many 9ths

5605 shillings. A. 191-101 days. A. 2¡1=211 quarts.

↑ XXXVI. To reduce a Fraction to its lowest Terms. Q. When an apple is divided into 4 parts, 2 parts, or 2, ar 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.

Q. Now, if we take 2, and multiply both the 2 and 4 by 2, we obtain ; what, then, is equal to? A., or . Q. Now it is plain that the reverse of this must be true; for, if we divide both the 4 and 8 in by 2, we obtain 2, and, di viding the 2 and 4 in by 2, we have ; what, then, may be inferred from these remarks respecting multiplying or divid ing both the numerator and denominator of the same fraction? A. That they may both be multiplied, or divided, by the same number, without altering the value of the fraction.

Q. What are the numerator and denominator of the same fraction called? A. The terms of the fraction.

Q. What is the process of changing into its equal♣ call ed? A. Reducing the fraction to its lowest terms.

Operation by Slate illustrated.

1. One minute is of an hour, and 15 minutes are 8. what part of an hour wiii

make, reduced to its lowest terms? How do you get the in this example ? A. By dividing 15 and 60, each, by 5. How do you get the? A. By dividing 3 and 12, each, by 3. How do you know that †

is reduced to its lowest terms? A. Because there is no num ber greater than 1 that will divide both the terms of without remainder.

From these illustrations we derive the following

RULE.

I. How do you proceed to reduce a fraction to its lowest terms? A. Divide both the terms of the fraction by any num ber that will divide them without a remainder, and the quo tients again in the same manner.

II. When is the fraction said to be reduced to its lowest terms? A. When there is no number greater than 1 tnat will divide the terms without a remainder.

More Exercises for the Slate.

of a barrel to its lowest terms.

2. Reduce
3. Reduce of a hogshead to its lowest terms
4. Reduce of a tun to its lowest terms.
of a foot to its lowest terms.
144

5. Reduce

6. Reduce

7. Reduce

90

↑ XXXVIII.

of a gallon to its lowest terms.

1. If 1 apple cost much is 2 times?

of an inch to its lowest terms.

A. 4.

A. Z.

A. 4.

A. Tz.

A..

A.

To multiply a Fraction by a Whole
Number.

of a cent, what will 2 apples cost? How

2. If a horse eat of a bushel of oats in one day; how many bushels will he eat in 2 days? In 3 days? How much is 2 times? 3 times ?

3. William has of a melon, and Thomas 2 times as much; what is Thomas' part? How much is 2 times ? 2 times ? 2 times 3 times ? 6 times?

Q. From these examples, what effect does multiplying the numerator by any number appear to have on the value of the fraction, if the denominator remain the same? A. It multiplies the value by that number.

Q. 2 times is = 2: but, if we divide the denominator 4 (in ) by 2, we obtain ; what effect, then, does dividing the denominator by any number have on the value of a fraction, if the numerator remain the same? A. It multiplics the value by that number.