2. A bought a farm containing 173 acres at $43 per acre, B bought 165 acres at $37 per acre; how much more did A's farm cost than B's?

3. A has 349 sheep, B has three times as many — 708, and C has as many as A and B together; how many sheep has B and C each, and how many have they all?

4. Two persons start from the same point, and travel in opposite directions; one travels 28 miles a day, and the other 35 miles. How far apart will they be in 19 days?

5. A drover bought 135 head of cattle at $27 a head, and 78 head at $43 a head, and sold the whole lot at $35 a head; what was his entire profit or loss?

6. A man's yearly income is $2375; he pays for houserent $437, his other expenses amount to 4 times as much $205; how much does he save yearly?

7. Multiply 753-(84+46) by (32X43)-(304-96).

8. A is worth $5250, B is worth $375 less than A, and C is worth as much as A and B together - $3107; how much is C worth?

9. If a cow cost $35, a horse 5 times as much, and a farm 10 times as much as the cow and horse together, – $309; how much more will the farm cost than 9 cows and 7 horses, at the same rate?

10. In an army of 17085 men, there were 1673 men killed in battle, and 4 times as many wounded, less the number that were taken prisoners, which was 1941; how many men remained in the army, and how many were wounded?

11. I exchanged a house worth $2160 and a store worth $2875, for 160 acres of land worth $43 per acre; how much do I still owe?

12. (352+552-651) X (332—32')=?

CHAPTER VI.

DIVISION OF SIMPLE WHOLE NUMBERS.

Art. 55. Division is the process of finding how many times one number contains another, or of finding one of the equal parts of a number.

The terms of division are, dividend, divisor, quotient, and remainder.

The dividend is the number to be divided.

The divisor is the number by which we divide.

The quotient is the number which shows how many times the divisor is contained in the dividend.

The remainder is the excess of the dividend over as many times the divisor as there are whole units in the quotient. Thus, 3 is contained in 14 four times and 2 remain. In this case, because the remainder 2 is two-thirds of the divisor 3, the true quotient is four and two-thirds. The remainder expresses the same kind of quantity as the dividend, because it is a part of the dividend.

While dividing, we must consider divisor and dividend as like numbers, because the divisor is, for the time being, considered as one of the parts of the dividend. Also, the quotient must, in dividing, be considered an abstract number, because it merely shows the number of times the dividend contains the divisor.

ILLUSTRATION.-If 3 persons share equally $15, each receives as many dollars as 3 is contained times in 15. Since 3 is contained in 15 five times, each person receives $5. In this example 3 persons and 15 dollars are not like numbers, but, in dividing, we make them like numbers by making both abstract. Also, the quotient, 5 times, is abstract and we reason from it to the concrete number, 5 dollars.

Art. 56. As multiplication can be illustrated by addition, so division can be illustrated by subtraction. Thus, we can illustrate the fact that 27 contains 9 three times, by subtracting 9 successively from 27 till it is exhausted. We find that three subtractions of 9 exhaust 27.

Art. 57. The signs of division are÷,—, and). Thus, 486-8, 48-8, 6)48-8, denotes that 48 is to be divided by 6, and that the quotient is 8; and is read 48 divided by 6 equals 8.

Art. 58. The equal parts into which a quantity may be divided have names which correspond with their number.

A half of a quantity is one of two equal parts which make that quantity. Hence a quantity is composed of 2 halves of itself.

A third of a quantity is one of three equal parts which make that quantity. Hence a quantity is equal to 3 thirds of itself.

A fourth of a quantity is one of four equal parts which make that quantity. Hence a quantity is equal to 4 fourths of itself.

In this manner other parts are named, thus:-fifths, sixths, sevenths, eighths, ninths, tenths, elevenths, twelfths, thirteenths, &c.; twentieths, twenty-firsts, twenty-seconds, twenty-thirds, &c.; one hundredths, one-hundred-and-firsts, one-hundred-and-seconds, &c.

Art. 59.

:

When the dividend is written over the divisor, the number thus formed may express four things, viz. :1ST. The division of the dividend into as many equal parts as there are units in the divisor. Thus, 15 may express the divi

sion of 15 into 3 equal parts.

2D. The division of the dividend into equal parts, each of which has the same value as the divisor. Thus, 15 may express the division of 15 into threes.

3D. One of those equal parts of the dividend whose name is represented by the divisor. Thus, may express one-third

4TH. As many of those equal parts of a unit whose name is represented by the divisor, as there are units in the dividend. Thus, may express fifteen thirds of 1.

15

3

In the last case the expression is a fraction, because it expresses a certain number of the equal parts of a unit.

Art. 60. The number below the line of a fraction is called the denominator, because it denominates, or names the parts into which the unit is divided.

The number above the line of a fraction is called the numerator, because it numerates, or states the number of the parts expressed by the fraction

Art. 61. To read a fraction, read first the number in the numerator, then the parts of a unit indicated by the denominator. Thus, is read two-thirds.

EXERCISES.

8

8

8

13

15

1. Read :::45:37:47:11:15:11:15 18:14. 2. Read 18: 1:15:55:11:13:38:14:43

Read101 86

6

23

95

99

56

12 30

3. Read 47: 182: 103:204: 188: 287: 365: 637.

17
406 507

Art. 62. The method of dividing one number by another is based on the following principles :

I. The number of times that the whole dividend contains the divisor is equal to the number of times that all the parts of the dividend contain the divisor.

II. Any excess, occurring from one part of the dividend's more than containing the divisor, should be united with the rest of the dividend to form a new part, in order to find how many times it contains the divisor.

A final remainder, being smaller than the divisor, and not containing it, can only be represented as divided. Hence it should be written with the divisor under it in the form of a fraction.

Ex. 1. Divide 789 by 5.

Ans. 157.

WRITTEN PROCESS.

Divisor. Dividend. 5)789

Quotient. 157

EXPLANATION.

5 is contained in 7 hundreds 1 hundred times, with a remainder 2 hundred. Unite the 2 hundreds with the 8 tens, making 28 tens; 5 is contained in 28 tens 5 tens times, with a remainder of 3 tens. Unite the 3 tens with the 9 units, making 39 units 5 is contained in 39 units 7 times, with a remainder 4. Since 4 does not contain 5, it must be represented as divided into five equal parts, each part being of 4, or 3. Hence the complete quotient is 157.

Ex. 2. How many times is 43 contained in 817254?

WRITTEN PROCESS. 43)817254 1900539 43

387

387

254

215

39

Ans. 1900539.

EXPLANATION.

If we do not know how many times 43 is contained in 81, we must make a trial with the left-hand figure, 4, of 43, and the left-hand figure, 8, of 81. Thus:-4 is contained in 8 twice; but, on trying 2 times 43, we find it 86, which is more than 81. Hence we try once 43, putting one in the quotient, and then multiply the divisor by this quotient figure; once 43 is 43, which subtracted from 81, leaves 38; to this remainder we annex or bring down 7, the next figure of the dividend, and thus form 387, the next partial dividend; 43 is contained in 387 nine times, and no remainder; we bring down 2 the next figure of the dividend and say "43 in 2 no times," and put 0 in the quotient, and bring down the next figure 5. Then "43 in 25 no times," and we put another 0 in the quotient, and bring down 4, making 254. Then, 43 in 254 5 times with a remainder 39. The remainder 39 is represented as divided, making the complete quotient 19005;}.

NOTE. When the division is performed mentally, and only the result written, the operation is called Short Division; but when all the work is written, it is called Long Division.

Rule.

Find how many times the divisor is contained in the fewest left-hand figures of the dividend that will contain it; this number of times is the first figure of the quotient.

Multiply the divisor by this quotient figure, and subtract the product from that part of the dividend which was used.

To the right of the remainder, if there is any, annex the next figure of the dividend.