the third 129 ft. long and 88 ft. wide; and the fourth 97 ft. long and 86 ft. wide. How many square feet are there in all of them?

27. Mr. Whitney sold 84 acres of land at $34.96 per acre, 138 acres at $27.58 per acre, and 427 acres at $49.64 per How much did he sell the whole for?

acre.

28. A city merchant went into the country to purchase flour. He was absent from the city 27 days, and his expenses while absent were $7.386 per day. He bought 175 bbl. of flour at $5.875 per bbl., 516 bbl. at $5.948 per bbl., 1386 bbl. at $6.11 per bbl., and 827 bbl. at $6.087 per bbl. It cost him $.634 per bbl. to get the flour transported to the city. He sold 697 bbl. of it at $7.114 per bbl., 824 bbl. at $7.213 per bbl., and the rest at $6.978 per bbl. Did he gain or lose by the adventure, and how much?

29. A merchant bought 49.5 cases of cassimere, each case containing 297 yd., at $1.1875 per yd. It cost him $.125 per case to have the cloth removed to his store, and $.045 per case to have it hoisted into his loft. One case of the cloth was stolen from him; he sold 23 cases at $1.423 per yd., and the remainder at $1.357 per yd., agreeing to deliver it at a railroad depot, 1 mile from his store. It cost him $.158 per case to have it carried to the depot. Did he gain or lose on the cloth, and how much?

SECTION VII.

DIVISION.

83. Definitions and Illustrations.

(a.) DIVISION IS A PROCESS BY WHICH WE ASCERTAIN

THE NUMBER OF PARTS OF A GIVEN SIZE INTO WHICH A GIVEN NUMBER MAY BE SEPARATED, OR BY WHICH WE

ASCERTAIN THE NUMBER

EACH PART OBTAINED BY

OF UNITS THERE WILL BE IN
DIVIDING A GIVEN NUMBER

INTO A GIVEN NUMBER OF EQUAL PARTS.

(b.) The following are questions in division:

4. How many apples at 3 cents apiece can be bought for 24 cents? 5 What is of 36?

6. If 8 apples cost 24 cents, how much will 1 apple cost?

(c.) In the first of the above questions, we are required to find how many 6's, or parts of 6 each, there are in 42; in the second, how many 9's, or parts of 9 each, there are in 54; in the third, how many 5's, or parts of 5 each, there are in 35; in the fourth, how many 3's, or parts of 3 each, there are in 24; while in the fifth we are required to find how many units there will be in each part obtained by dividing 36 into 4 equal parts; and in the sixth, how many units there will be in each part obtained by dividing 24 into 8 equal parts.

(d.) This shows that there are two classes of questions in division, viz.: one class in which, knowing the number to be divided, and the num ber of units which each part is to contain, we are required to find the number of parts; and one in which, knowing the number to be divided, and the number of parts into which it is to be divided, we are required to find how many units there will be in each part; i. e., we are required to find a fractional part of a number. Both classes of questions may be solved by the same numerical process, though in their solution they require different reasoning processes.

(e.) The number to be divided is called the dividend. In the first class of questions the number which shows the size of each part, and in the second class that which shows the number of parts, is called the divisor. The result is called the quotient.

Illustrations. In the first of the above questions, 42 is the dividend, 6 is the divisor, and the answer, 7, is the quotient.

In the second, 54 is the dividend, 9 is the divisor, and the answer, 6, is the quotient.

In the fifth, 36 is the dividend, 4 is the divisor, and the answer, 9, is the quotient.

(f) The fourth example would be solved thus:

If for 3 cents one apple can be bought, for 24 cents as many apples

can be bought as there are times 3 cents in 24 cents, which are 8 times. Therefore, 8 apples can be bought for 24 cents, when 1 apple can le bought for 3 cents. Here, 24 is the dividend, 3 is the divisor, and the answer, 8, is the quotient. The divisor and dividend are of the same denomination, and the quotient is the number of times the divisor is contained in the dividend, or the number of parts equal to the divisor which the dividend equals.

(g.) The sixth example would be solved thus :

If 8 apples cost 24 cents, 1 apple will cost of 24 cents, which is 3 cents. Hence, 1 apple will cost 3 cents, if 8 cost 24 cents.

Here, 24 is the dividend, 8 is the divisor, and the answer, 3, is the quotient. The dividend and quotient are of the same denomination, and the divisor shows the number of times the quotient is taken in the dividend, or the number of parts equal to the quotient which the dividend equals.

(h.) To examine more fully the nature of the processes, let us see by what methods the answers to the fourth and sixth questions could be obtained.

(i.) It is obvious that, to determine the answer to the fourth question, we must find how many parts of 3 cents each there are in 24 cents; for each such part is the price of 1 apple.

We can do this by counting 24 cents into piles of 3 cents each, and then counting the number of piles; - by finding how many threes must be added to make 24; - or by finding how many times 3 equal 24, by our knowledge of multiplication. This last process is division.

(j.) To determine the answer to the sixth question, we must find how many cents there will be in each part obtained by separating 24 cents into 8 equal parts.

We can do this by laying out 24 cents into 8 equal piles, and then counting the number of cents in each pile ; -or by finding, by our knowledge of multiplication, what number must be taken 8 times to equal 24; or, which will give the same numerical answer, by finding how many times 8 equal 24. The last process is division.

NOTE. It will be seen, that in the first class of questions, the divisor and dividend are of the same denomination, and that the quotient expresses the number of times the divisor is contained in the dividend, or the number of parts equal to the divisor which must be taken to produce the dividend; while in the second class, the divisor expresses the number of parts into which the dividend is to be divided, and the quotient expresses the number of units in each part; thus making the dividend and quotient of the same denomination.

(k.) Practically, division is the reverse of multiplication. In the

latter, the factors are given, and we are required to find the product while in the former, one of the factors and their product are given, and we are required to find the other; or, when the dividend will not exactly contain the divisor, one factor and the product of the two plus the remainder are given, and we are required to find the other factor and the remainder.

(1.) The divisor is always the given factor, the quotient is the required factor, and the dividend is the product, or the product plus the remainder. The remainder is always less than the divisor.

(m.) The following examples illustrate this:

1. "3 times 8," is a question in multiplication. The factors 3 and 8 are given, and the product, 24, is required.

2. "How many times 8 = 24," or " 24 ÷ 8?" are questions in division, in which the factor 8, and the product 24, are given, and the missing factor, 3, is required.

3. "of 24, or 3 times what number = 24?" are questions in division, in which the factor 3, and the product 24, are given, and the missing factor is required.

4. "In 8 times 7, plus 5," the factors 8 and 7 are given, and their product, plus 5, which is 61, is required.

5. "In 617," or "61 = how many times 7," the factor 7, and the sum of the product, and remainder, which is 61, are given, and the other factor and remainder are required.

6. “In 4 of 61," or "7 times what number = 61," the factor 7, and the sum of the product, plus the remainder, are given, and the other factor and remainder are required.

84. Methods of Proof.

From the preceding illustrations it is evident,

First. That, where there is no remainder, the divisor multiplied by the quotient must produce the dividend; and that the dividend divided by the quotient must produce the divisor.

Second. That if, when there is a remainder, the divisor and quotient be multiplied together, and the remainder be added to their product, the result will equal the dividend.

Third. That if the remainder be subtracted from the dividend, and the remainder thus obtained be divided by the divisor, the result will equal the quotient; or, if it be divided by the quotient, the result will equal the divisor.

NOTE. The remain ler should always be less than the divisor; for,

if it is not, the dividend will contain the divisor more times than is indi cated by the quotient figure.

85. Examples. Quotient a single Figure.

(a.) How many oranges at 7 cents apiece can be bought for 61 cents?

Reasoning Process. -If for 7 cents 1 orange can be bought, as many

oranges can be bought for 61 cents as there are times 7 in 61.

Explanation. To perform the necessary division, we observe that as 568 times 7, 61 must equal 8 times 7, with a remainder of 5; or, since 5 ÷ 7 = 4, 61 = 8 times 7.

Hence, the quotient is 8, and the remainder is 7, or the complete quotient is 85, which shows that 8 oranges can be bought, leaving 5 cents unused, or that 85 oranges can be bought with all the money.

First Proof. See if the price of 8 oranges at 7 cents apiece, added to 5 cents, will make 61 cents.

Second Proof. Considering that 85 oranges are bought, see if they will cost 61 cents at 7 cents apiece.

(b.) The above are proofs of the correctness of the entire work; the following only test the correctness of the division.

Third Proof. 8 times 756, and 5 added = 61 = dividend. Fourth Proof. 8 times 7 56; and 56 from 61 leaves 5 remainder.

Fifth Proof. 5 from 61 = 56, and 56 ÷ 7

(c.) The work may be written thus:

Dividend.

= 8= quotient.

(d.) Or, by expressing the division of the remainder, by placing it in the form of a fraction, we have,

In the first form, the remainder is undivided, and is of the same denomination as the dividend. It is placed opposite the dividend, with the minus sign between, to indicate that all the dividend except that