DIVISION.

(33.) DIVISION consists in finding how many times one number contains another, or what part one number is of another.

The number to be divided is called the dividend, the dividing number the divisor, and the number or part found the quotient.

If we divide 15 by 5, the quotient will be 3, because 5 is contained in 15, 3 times.

What is the quotient of 6 divided by 3?
What is the quotient of 20 divided by 4?
What is the quotient of 56 divided by 8 ?

Of 12 divided by 4?
Of 36 divided by 9?
Of 63 divided by 7?

One-half is one of the two equal parts of any quantity; two-thirds are two of the three equal parts of any quantity,

A less Number divided by a greater.

(34.) The Quotient of a less number divided by a greater is the part that the less is of the greater; and is denoted by the less over the greater, with a line between them.

1 divided by 2 is, one-half, because 1 is one-half of 2; 2 divided by 3 is, two-thirds, because 2 is two-thirds of 3.

How much is 1 divided by 3? and why? How much is 1 divided by 4 and why? How much is 3 divided by 4? and why? How much is 1 divided by 5? and why? How much is 2 divided by 7? and why?

The reciprocal of a number is a unit divided by that number; thus the reciprocal of 2 is, and the reciprocal of 3 is 1.

What is the reciprocal of 4? Of 5? Of 6? Of7?

Relation of the Quotient to the Dividend.

(35.) The Quotient is always that part of the dividend which is denoted by the reciprocal of the divisor.

Thus the quotient of 15 divided by 5 is 3, and 3 is of 15. The quotient of 2 divided by 3, is

(34), and two-thirds

How would you find of any given number? of 1 is what part of 2?. of 1 is what part of 4? of 1 is what part of 7?

Remainder in Division.

(36.) A remainder, in Division, is an overplus of the dividend above the repetitions of the divisor contained in it; and may be divided separately, to complete the quotient (34).

Thus 5 is contained in 17, 3 times with 2 over; this 2 divided by 5 gives, which, annexed to 3, makes the complete quotient, 33, three and two-fifths.

What is the quotient of 17 divided by 2?
What is the quotient of 35 divided by 4?
What is the quotient of 50 divided by 6?

Of 20 divided by 3?
Of 47 divided by 5?
Of 65 divided by 72

Multiplication and Division.

(37.) Multiplication and Division are the reverse of each other in Multiplication two factors are given to find their product; in Division a product and one of its factors are given to find the other factor.

The product being 60, and one factor 5, what is the other factor? The product being 72, and one factor 8, what is the other factor? The product being 85, and one factor 9, what is the other factor?

The sign, called by, placed between two numbers, signifies that the first number is to be divided by the second; thus 369, 36 by 9, signifies that 36 is to be divided by 9.

What is the quotient of 568? Of 639? Of 77÷11?
What is the quotient of 648? Of 84-7? Of 100 ÷ 12?

Division is also denoted by the dividend over the divisor, with a line between them.

Thus 36 denotes 36 divided by 9.

Constant Quotient.

(38.) The quotient evidently remains the same, when the dividend and divisor are both multiplied or both divided by the same number.

Thus 3 is contained in 15 just as often as 4 times 3 is contained in 4 times 15.

RULE VII.

(39.) To divide by a Number not exceeding 12, or 12 with O's annexed.

1. Take figures enough in the left of the dividend to contain the divisor, and set down the number of times the divisor goes therein, noticing the overplus, if any.

2. Take the next figure of the dividend, with the preceding overplus, if any, prefixed, and set the number of times the divisor goes therein on the right of the first quotient; if the divisor, will not go therein, set down 0, and include the next figure in dividing; and so on.

3. Ciphers in the right of the divisor are omitted in dividing; but as many figures must be omitted in the right of the dividend, and annexed to the remainder: if there be no other remainder, these figures will form the remainder.

4. Under the remainder, if any, set the given divisor, to complete the quotient.

EXAMPLE I.

To find how many times 9 is contained in 23472.

9)23472
2608

We say, 9 in 23, twice, and 5 over; prefixing this 5 to the 4, we say, 9 in 54, 6 times; 9 in 7, 0 time; 9 in 72, 8 times.

The overplus of any particular place in the dividend, is so many tens in the next place on the right (10); and is made tens to the next figure by prefixing it to that figure.

EXAMPLE II.

To find how many times 120 is contained in 13127.

120) 13127
109130

In dividing, we omit the 0 in the right of the divisor, and the 7 in the right of the dividend.-12 in 13 goes once, and 1 over; prefixing this 1 to the next figure, we say, 12 in 11, O time; including the next figure, we say, 12 in 112, 9 times, and 4 over; to this 4 we annex the 7 omitted, and get the remainder 47, under which we set the whole given divisor 120.

The quotient thus found is evidently the same as if we had used the entire divisor 120, and taken one figure more, each time, of the dividend.

When the Divisor is 10, 100, or 1000, &c.

(40.) A number is divided by 10, 100, or 1000, &c., by cutting off from the right of the dividend as many figures as there are O's in the divisor.

The other figures of the dividend will be the quotient, and those cut off, the remainder.

Thus 37560-100 gives the quotient 375, and the rem. 60.

The Operation Proved.

(41.) Division may be proved by multiplying the divisor and quotient together, and adding the remainder, if any, to the product; the result must be equal to the dividend

EXERCISES.

1. How many barrels of apples, at 2 dollars a barrel, may be bought for 150 dollars?

The number of barrels that may be bought is the number of times that 2 is contained in 150. Ans. 75 barrels.

2. How many yards of broadcloth, at 3 dollars a yard, may be purchased for 387 dollars? Ans. 129 yards. 3. How many cords of wood, at the rate of 4 dollars per cord, may be purchased for 621 dollars?

4. How many superfine beaver hats, at may be purchased for 3700 dollars?

Ans. 155 cords.

5 dollars apiece,

Ans. 740 hats.

5. How many dozen of shoes, at the rate of 6 dollars per dozen, may be purchased for 775 dollars? Ans. 129 dozen. 6. There being 7 days in a week, it is required to find how many weeks there are in 728 days. Ans. 104 weeks.

7. If one box will hold 80 pair of shoes, how many of such boxes will be required to hold 1840 pair? Ans. 23 boxes. 8. At the rate of 90 dollars per acre, how many acres of land could be bought for 5237 dollars? Ans. 5817 acres.

9. At the rate of 10 dollars per barrel, how many barrels of flour could be purchased for 1890 dollars?

Ans. 189 barrels.

10. At 11 dollars each, per month, how many laborers could be hired a month for 2530 dollars? Ans. 230 laborers.

11. If 2 acres of ground sell for 365 dollars, what is the price per acre?

If 2 acres bring 365 dollars, 1 acre brings of 365 dollars. Ans. 182 dollars.