From the preceding demonstration it follows that,

§ 120. In fractional, as in integral, multiplication, the pro duct remains the same, when the multiplicand and multiplier are taken, the one for the other. Thus of 3 of §.

11. What should be paid for 2 of a yard of linen, at the rate of of a dollar per yard?

It is plain that of a yard will cost of of a dollar.

of

, the same as X, is a compound fraction, to be reduced to a simple fraction. (§ 118). Ans. of a dollar.

apples, if the

12. What should be paid for of a barrel of whole barrel be worth 1 of a dollar? Ans. of a dollar. 13. A person owning of a tract of land, sells of his share to A; what part of the whole tract does A purchase? Ans. of the whole.

14. What should be paid for 1⁄2 of 7 of a pound of tea, at the rate of 1% of a dollar per pound? Ans. of a dollar.

15. A gentleman owning of a ship, sells of his share to A, and the rest of it to B. What part of the whole ship does he sell to each? Ans. To A, and to B of the whole. 16. Bought of an acre of ground at the rate of 18 dollars per acre; required the sum to be paid for it.

Ans. 11 dollars. 17. Sold 25 bushels of clover seed at 74 dollars per bushel, and 3 bushels at 7 dollars per bushel; what did the whole amount to? Ans. 205 dollars. 18. Find the entire cost of of a pound of pepper at of a dollar a pound, & of a hundred weight of flour at 2 dollars a hundred, and 2 yards of cloth at 7 dollars a yard. Ans. 1615 dollars.

In multiplying two or more fractions together, equal factors may be cancelled in the resulting numerator and denominator, without affecting the value of the product. (§ 92, § 69).

19. Reduce of of of to a simple fraction.

13

20. Reduce of 3 of 3 of 71⁄2 to a simple expression. Ans. 11. A factor of a composite number is canceled by substituting the other factor of that number for the number itself. (§ 66).

24. Having purchased 350 bushels of wheat at of a dollar a bushel, and sold of the quantity at 1 dollar a bushel, what would I gain or lose on the whole by selling the remainder at dollar a bushel? Ans. Gain 14 dollars. 25. In how many days ought one man to accomplish an undertaking which 17 men could perform in 133 days? Ans. 2273 days. 26. A farmer bought of a tract of land which contained 735 acres, and sold to his neighbor of his purchase. part of the whole tract, and how many acres, did he sell?

27. Bought of A 3

What

Ans. of the tract; 294 acres. cords of wood, at 2 dollars per cord; of B 73 cords, at 15 dollars per.cord; and of C 10 cords, at 2 dollars per cord. What was the whole quantity of wood purchased, and the whole sum paid for it?

Ans. 207 cords; 421 dollars. 28. If 3 masons can build a wall of a certain length, height, and thickness, in 13 days, in what time ought one mason to build a wall of the same height and thickness, but 24 times as long? Ans. 911 days.

He

29. A speculator bought of A 189 acres of land at 10 dollars an acre, and of B 250 acres at 13 dollars an acre. sold of the first tract at 181 dollars, and of the second at 19 dollars an acre; what would he make, on the whole, by selling the remainder of both tracts at 20 dollars an acre? Ans. 335213 dollars.

DIVISION OF FRACTIONS.

$121. Dividing by a fraction, as well as by an integer, consists in finding how many times the dividend contains the divisor, or what part the dividend is of the divisor.

Thus divided by gives the quotient 3, because 3 times; and divided by isof.

How many times dues

by? Of divided by

contains

gives the quotient, because

contain, or what is the quotient of § divided Of af divided by ?

What part is of, or what is the quotient of

divided by ? Of divided by ? Of divided by ? Of ÷? Of ÷Y?

§ 122. In fractional, as in integral, division, the dividend is a product given, and the divisor one of its factors given, to find the other factor.

§ 123. The reciprocal of a fraction is the fraction inverted ; and is equal to a unit divided by the fraction. (§ 50).

Thus the reciprocal of is; equal to 1 or

What is the reciprocal of

divided by .

? Of ? of? Of }} ?

The reciprocal of a mixed number is that of its equivalent improper fraction. Thus the reciprocal of 537 is.

What is the reciprocal of 24? Of 73? Of 103? Of 9,3?

Complex or Mixed Fractions.

§ 124. When the dividend or divisor is a fraction or a mixed number, the dividend placed over the divisor, with a line between, forms a complex or mixed fraction.

Thus, 25, 2-3, may be expressed, respectively, by the

These complex fractions may be read-numerator 3, denominator 5; numerator 2, denominator 3.

A complex, as well as a simple, fraction is equal to its numerator divided by its denominator. (§ 92).

RULE XXIII.

§ 125. To divide a fraction by a fraction.

1. Divide the numerator of the dividend by that of the divisor, and the denominator by the denominator; or

2. Multiply the dividend by the reciprocal of the divisor.

3. A fraction is divided by an integer, by dividing the numerator by the integer, or multiplying the denominator by the integer.

4. An integer is divided by a fraction, by multiplying the integer by the reciprocal of the fraction.

5. A mixed number may be used in division under the form of an improper fraction. A mixed number may also be divided by an integer, by dividing the integer and fraction in the mixed number, separately.

1. To divide 8 by %.

EXAMPLES.

Dividing numerator by numerator, and denominator by de. nominator,

Here we cannot divide numerator by numerator, and denominator by denominator, without remainders. Multiplying, therefore, the dividend by the reciprocal of the divisor,

we have=&XI=1}

3. To divide by 5.

Dividing the numerator of the fraction by the integer,
10-5

we have 15= 13 =13.

Or, multiplying the denominator by the integer,

10

we have ÷5=13x5=&f=15, as before.

4. To divide 123 by .

Maltiplying the integer by the reciprocal of the fraction, we have 123-1=123X=615-153.

5. To divide 163 by 5.

Reducing the mixed number to an improper fraction, we have 1633-5=43÷÷5=4%;'=32}}.

Otherwise. Dividing the integer and fraction in the mixed number, separately,

5)163
321

We say 5 in 16, 3 times and 1 over; 5 in 13, twice and 3 over: this 3 and the make 3, equal to ; then÷÷5, that is, of, is 11.

Dividing a fraction by a fraction is the reverse of multiplying a fraction by a fraction. (§122).

When the terms of the dividend cannot be divided by those of the divisor without a remainder, suppose each term of the former X by both terms of the latter; this will not alter the value of the dividend. (§93). Then the resulting numerator ÷ the numerator of the divisor, and the denominator by the denominator, gives, for the quotient, the dividend X the reciprocal of the divisor.

From the Rule, thus demonstrated, it follows that,

§ 126. Dividing by a Fraction is equivalent to multiplying the dividend by the reciprocal of the divisor.

In thus multiplying, equal factors may be canceled in the resulting numerator and denominator, (§ 92, § 69).

Thus ××, or simply, by only dividing 8 and 4 by 4. and 15 and 5 by 5.