15. What will be the cost of 2 pks. and 4 qts. of wheat, at $150 per bushel?

16. Supposing a meteor to appear so high in the heavens as to be visible at Boston, 71° 3', at the city of Washington, 77° 43', and at the Sandwich Islands, 155° W. longitude, and that its appearance at the city of Washington be at 7 minutes past 9 o'clock in the evening; what will be the hour and minute of its appearance at Boston and at the Sandwich Islands?

FRACTIONS.

¶ 43. We have seen, (¶ 17,) that numbers expressing whole things are called integers, or whole numbers; but that, in division, it is often necessary to divide or break a whole thing into parts, and that these parts are called fractions, or broken numbers.

It will be recollected, (T 14, ex. 11,) that when a thing or unit is divided into 3 parts, the parts or fractions are called thirds; when into four parts, fourths; when into six parts, sixths; that is, the fraction takes its name or denomination from the number of parts, into which the unit is divided. Thus, if the unit be divided into 16 parts, the parts are called sixteenths, and 5 of these parts would be 5 sixteenths, expressed thus, The number below the short line, (16,) as before taught, ( 17,) is called the denominator, because it gives the name or denomination to the parts; the number above the line is called the numerator, because it numbers the parts.

The denominator shows how many parts it takes to make a unit or whole thing; the numerator shows how many of these parts are expressed by the fraction.

1. If an orange be cut into 5 equal parts, by what fraction is 1 part expressed?

4 parts?

or a whole orange?

2 parts?

3 parts? 5 parts? how many parts make unity

2. If a pie be cut into 8 equal pieces, and 2 of these pieces be given to Harry, what will be his fraction of the pie? if 5 pieces be given to John, what will be his fraction? what fraction or part of the pie will be left?

It is important to bear in mind, that fractions arise from division, († 17,) and that the numerator may be considered a

dividend, and the denominator a divisor, and the value of the fraction is the quotient; thus, is the quotient of 1 (the numerator) divided by 2, (the denominator;) is the quotient arising from 1 divided by 4, and is 3 times as much, that is, 3 divided by 4; thus, one fourth part of 3 is the same as 3 fourths of 1.

Hence, in all cases, a fraction is always expressed by the sign of division.

3 is the dividend, or numerator.

& expresses the quotient, of which is the divisor, or denominator. 3. If 4 oranges be equally divided among 6 boys, what part of an orange is each boy's share?

A sixth part of 1 orange ist, and a sixth part of 4 oranges is 4 such pieces, t. Ans. of an orange.

4. If 3 apples be equally divided among 5 boys, what part of an apple is each boy's share? if 4 apples, what? if 2 apples, what? if 5 apples, what?

5. What is the quotient of 1 divided by 3?

6. What part of an orange is a third part of 2 oranges?

of 1 by 4?

of 2 by 4?

of 3 by 4?

of 2 by 3? of 5

by 7?

of 6 by 8?

of 4 by 5?

of 2 by 14?

of 3 oranges?

of 2?

† of 5 i

-one fourth of 2 oranges?

A Proper Fraction. Since the denominator shows the number of parts necessary to make a whole thing, or 1, it is plain, that, when the numerator is less than the denominator, the fraction is less than a unit, or whole thing; it is then called a proper fraction. Thus,,, &c. are proper fractions.

An Improper Fraction. When the numerator equals or exceeds the denominator, the fraction equals or exceeds unity, or 1, and is then called an improper fraction. Thus, &, †, †, L, are improper fractions.

A Mixed Number, as already shown, is one composed of a whole number and a fraction. Thus, 141, 137, &c. are mixed numbers.

7. A father bought 4 oranges, and cut each orange into 6 equal parts; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces; what was each one's fraction?

Was James's fraction proper, or improper? Why?
Was Nancy's fraction proper, or improper? Why?

To change an improper fraction to a whole or mixed number. T 44. It is evident, that every improper fraction must contain one or more whole ones, or integers..

1. How many whole apples are there in 4 halves (1) of

in

-in § of a yard? in?

an apple? 20 ?

in § ?

in ?

in 48? 2. How many yards in

in 120?

in 10? in 284?

of a yard?

in ?

in ??

-in 10?

.in

This finding how many integers, or whole things, are contained in any improper fraction, is called reducing an improper fraction to a whole or mixed number.

4. If I give 27 children of an orange each, how many oranges will it take? It will take 27; and it is evident, that

OPERATION.

4)27

Ans. 6 oranges.

dividing the numerator, 27, (= the number of parts contained in the fraction,) by the denominator, 4, (= the number of parts in 1 orange,) will give the number of whole oranges.

Hence, To reduce an improper fraction to a whole or mixed number,-RULE: Divide the numerator by the denominator; the quotient will be the whole or mixed number.

EXAMPLES FOR PRACTICE.

5. A man, spending † of a dollar a day, in 83 days would spend 8 of a dollar; how many dollars would that be? Ans. $13.

6. In 1417 of an hour, how many whole hours? The 60th part of an hour is 1 minute: therefore the question is evidently the same as if it had been, In 1417 minutes, how many hours? Ans. 233 hours.

7. In 9783 of a shilling, how many units or shillings? Ans. 730 shillings.

8. Reduce 146 to a whole or mixed number.

9. Reduce 28, 106, 178, 1788, 248, to whole or mixed numbers.

To reduce a whole or mixed number to an improper fraction.

¶ 45. We have seen, that an improper fraction may be changed to a whole or mixed number; and it is evident, that, by reversing the operation, a whole or mixed number may be changed to the form of an improper fraction.

1. In 2 whole apples, how many halves of an apple? Ans. 4 halves; that is, . In 3 apples, how many halves? in 4 apples? in 6 apples? in 10 apples? in 24? in 60? in 170 in 492 ?

Reduce 23 yards to

2. Reduce 2 yards to thirds. Ans. §. thirds. Ans. . Reduce 3 yards to thirds. 5 yards.

3 yards.

yards.

3 yards.

5 yards.

63

72 bushels.

of a dollar?

3. Reduce 2 bushels to fourths.

61 bushels.

4. In 165 dollars, how many

22.bu, 6 bushels. 252 bushels.

make 1 dollar: if, therefore, we multip y 16 by 12, that is, multiply the whole number by the denomi sator, the product will be the number of 12ths in 16 dollars: 16 x 12 192, and this, increased by the numerator of the fraction, (5,) evidently gives the whole number of 12ths; that is, 197 of a dollar, Answer.

OPERATION.

16 dollars.

12

192 12ths in 16 dollars, or the whole number.
512ths contained in the fraction.

197= 197, the answer.

Hence, To reduce a mixed number to an improper fraction,RULE: Multiply the whole number by the denominator of the fraction, to the product add the numerator, and write the result over the denominator.

EXAMPLES FOR PRACTICE.

5. What is the improper fraction equivalent to 237 hours? Ans. 147 of an hour.

6. Reduce 730 shillings to 12ths. As of a shilling is equal to 1 penny, the question is evidently the same as, In 730 s. 3 d., how many pence?

Ans. 873 of a shilling; that is, 8763 pence.

7. Reduce 118, 1728, 8750, 478%, and 725 to improper fractions.

8. In 156 days, how many 24ths of a day?

Ans. 37013761 hours.

9. In 342 gallons, how many 4ths of a gallon?

Ans. 1371 of a gallon 1371 quarts.

To reduce a fraction to its lowest or most simple terms.

146. The numerator and the denominator, taken together, are called the terms of the fraction.

If of an apple be divided into 2 equal parts, it becomes 4. The effect on the fraction is evidently the same as if we had multiplied both of its terms by 2. In either case, the parts are made 2 times as MANY as they were before; but they are only HALF AS LARGE; for it will take 2 times as many fourths to make a whole one as it will take halves; and hence it is that is the same in value or quantity as

is 2 parts; and if each of these parts be again divided into 2 equal parts, that is, if both terms of the fraction be multiplied by 2, it becomes. Hence, ==, and the reverse of this is evidently true, that

It follows therefore, by multiplying or dividing both terms of he fraction by the same number, we change its terms without altering its value.

Thus, if we reverse the above operation, and divide both terms of the fraction by 2, we obtain its equal, 4; dividing again by 2, we obtain, which is the most simple form of the fraction, because the terms are the least possible by which the fraction can be expressed.

The process of changing into its equalis called reducing the fraction to its lowest terms. It consists in dividing both terms of the fraction by any number which will divide them both without a remainder, and the quotient thence arising in the same manner, and so on, till it appears that no number greater than 1 will again divide them.

A number, which will divide two or more numbers without a remainder, is called a common divisor, or common measure of those numbers. The greatest number that will do this is called the greatest common divisor,