How many times may 5 be subtracted from 20? How many times may 6 be subtracted from 24? 7 from 35? 8 from 48?

Multiplication and Division.

$49. Multiplication and Division are the reverse of each other.

In Multiplication, two numbers or 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 15, and one factor 3, what is the other factor? The product being 30, and one factor 5, what is the other factor? The product being 36, and one factor 9, what is the other factor? The product being 63, and one factor 7, what is the other factor?

Reciprocal of a Number.

§ 50. The reciprocal of a number is a unit or 1 divided by that number.

Thus the reciprocal of 2 is, and of 3 is §.

What is the reciprocal of 4? Of 5? Of 6? Of 10? Of 20?

The Quotient as a Part of the Dividend.

§ 51. The Quotient is always such a part of the dividend as is expressed by the reciprocal of the divisor.

Thus 15 divided by 5 gives 3, and 3 is of 15.

Again, if we divide 2 by 3, the quotient will be 3 (§ 47); and since two thirds of any quantity is one third of two such quantities, is equal to of 2; such a part of the dividend 2 as is expressed by the reciprocal of the divisor 3.

of 1 cent is what part of 3 cents ?
of 1 pint is what part of 2 pints?
of 1 mile is what part of 5 miles?

of 5 cents is what part of 1 cent?
of 7 pints is what part of 1 pint?
of 8 miles is what part of 1 mile?

How would you find of any number

3 of 1 is what part of 3?

4

2 of 1 is what part of 2 ?

8

of 1 is what part of 5? of 5 is what part of 1? of 7 is what part of 1? of 8 is what part of 1?

of any number?

any number? of any number? of any number ?

Sign of Division

of

$ 52. The sign÷, called by, placed between two numbers, signifies that the first of them is to be divided by the second. Thus 36-9, 36 by 9, signifies that 36 is to be divided by 9

Division is also denoted by the dividend over the divisor with

a line between them.

Thus denotes the same as 63÷9.

Quotient of Concrete Numbers.

$53. When the dividend and divisor are similar concrete numbers, the quotient is the number of times the dividend contains the divisor, or the part the dividend is of the divisor. Thus 12 cents÷3 cents gives 4; and 12 cents 13 cents gives (§ 47).

5 pounds in 30 pounds? 77 yards? 12 dollars in 96 dollars? What part is ounces of 20 ounces

How many times 4 miles in 12 miles?
6 inches in 42 inches? 7 yards in
What part is 3 days of 7 days?
What part is 20 feet of 49 feet?

$ 54. When the dividend and divisor are dissimilar concrete numbers, the quotient is such a part of the dividend as is expressed by the reciprocal of the divisor taken abstractly.

For example, if 5 pencils cost 20 cents, one pencil will cost 20 cents-5; that is, of 20 cents, which is 4 cents.

If 3 slates cost 36 cents, what will one slate cost? If 4 hats cost 16 dollars, what will one hat cost? If in 9 hours a stage runs 54 miles, at what rate does it run per hour?

Remainder in Division.

$55. A remainder, in Division, is an overplus or excess of the dividend above so many times the divisor as it is contained in the dividend.

Thus 5 is contained in 17, 3 times, with 2 over, since 3 times 5 is 15; then 2 is the remainder of the dividend.

If the divisor be 6, and the dividend 27, what will the quotient and the remainder be? If the divisor be 8, and the dividend 45? If the divisor be 9, and the dividend 70 ?

§ 58. The remainder divided by the divisor, and so annexed to the quotient, completes the quotient.

Thus 175 gives quotient 3, and remainder 2. This 2-5 gives (47); the complete quotient is then 33, three and twofifths.

In like manner find the quotient of 9-2. Of 134. Of 21÷÷÷5. Of 27-6. Of 30-7. Of 41-8. Of 100-9.

Constant Quotient.

§ 57. The quotient of two numbers remains the same, when those numbers are both multiplied, or both divided, by the same number.

For example, 6-2 gives 3, and 5 times 6÷÷÷5 times 2, that is, 30-10, also gives 3. By reversing the process, we find 30-10 equal to of 30 of 10.

RULE VII.

§ 58. To Divide by a number not exceeding 12; or by such Number with Os 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, noting the excess, if any.

2. Take the next figure of the dividend, with the preceding excess, if any, prefixed, and set the number of times the divisor is found therein on the right of the first quotient; if the divisor will not go therein, put a 0 in the quotient, 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. When the divisor is 10 or 100, &c., take for the remainder as many figures from the right of the dividend as there are Os in the right of the divisor, and the other figures of the dividend for the quotient.

5. Under the remainder, if any, set the given divisor, and annex the part so found to the quotient.

EXAMPLES.

1. To divide 7805 by 3.

3)7805

26013

We say, 3 in 7, twice, and 1 over; prefixing the 1 to the next figure 8 of the dividend, we have 18; 3 in 18, 6 times; 3 in 0, 0 time; 3 in 5, once, and 2 over. This excess 2 is the remainder,-under which setting the divisor 3, we have to annex to the quotient (§ 56).

The quotient 2601 shows that the dividend 7805 contains the divisor 3, 2601 times, and of the divisor, besides.

The quotient is also of 7805 (§ 51.)

2. To divide 13127 by 120.

Omitting the 0 in the right of the divisor, and the 7 in the right of the dividend, we say,

12 in 13 once, and 1 over; prefixing the 1 over to the next figure of the dividend, we say 12 in 11, 0 time; including the next figure 2 of the dividend, we say 12 in 112, 9 times, and 4

over.

47

Annexing the 7, omitted, to the 4, we have the remainder 47, under which setting the given divisor 120, we have to annex to the quotient.

120

3. To divide 14723 by 100, we take the two figures 23 from the right of the dividend, for the remainder, and the other figures 147 for the quotient.

Hence the complete quotient is 147,23

The figures of the dividend first taken in dividing, have a local value 10 times, or 100 times, &c., their simple value, according as one, or two, &c., figures follow them on the right (§ 16); and the first quotient figure must therefore have 10 times, or 100 times, &c., its simple value. The proper local value is assigned to the quotient figure, by the succeeding quotient figures, these being always just as many as the succeeding figures of the dividend. In like manner each quotient figure receives its proper value.

The excess belonging to any particular place in the dividend, is so many tens in the next place on the right (§ 11); and is made tens to the next figure by prefixing it to that figure.

In the second example, we divided 12 in 1312, instead of 120 in 13127. But 120 is 10 times 12, and 13127 is 13120+7, or 10 times 1312,7; and 12 in 1312 gives the same quotient as 10 times 12 in 10 times 1312 (§ 57).

In the same way it may be shown that any number of Os may be omitted in the right of the divisor, if an equal number of figures be omitted in the right of the dividend.

The Operation Proved.

§ 59. Division may be verified or 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 per barrel, may De bought for 150 dollars?

The number of barrels that may be bought, is the number of times that 2 dollars is contained in 150 dollars (§ 53).

Ans. 75 barrels.

2. At 3 dollars per yard, how many yards of broadcloth may be purchased for 387 dollars Ans. 129 yards. 3. How many cords of wood, at 4 dollars per cord, may be purchased for 600 dollars? Ans. 150 cords.

4. At 5 dollars apiece, how many superfine beaver hats may be purchased for 3700 dollars? Ans. 740 hats.

5. How many dozen of shoes, at 6 dollars per dozen, may be purchased for 750 dollar? Ans. 125 dozen. 6. There being 7 days in a week, it is required to find how many weeks there are in 728 days! A s. 104 weeks.

7. If one box will hold 80 pair of shoes, how many of such boxes will be required to coin 1840 pair? Ans. 23 boxes. 8. At the rate of 900 dollars each, how many dwelling houses could be built for 11700 dollars? Ans: 13 houses. 9. At 11 dollars each per month, how many laborers could be hired a month for 2530 dollars? Ans. 230 laborers. 10. At 120 dollars apiece, how many fine horses could I purchase for the sum of 4200 dollars? Ans. 35 horses.

11. What quantity of flour could be bought for 3 dollars, when the price is 5 dollars per barrel?

One dollar would buy of a barrel; hence 3 dollars would buy 3 of a barrel.

Or, 3 dollars would buy the same part of a barrel that 3 dollars is of 5 dollars: 3 dollars is of 5 dollars (§ 47).

Ans. of a barrel.

12. At the rate of 15 dollars per ton, what quantity of hay could be bought for 7 dollars? Ans. of a ton. 13. If land sell at the rate of 50 dollars per acre, what quantity of land could be purchased for 17 dollars?

Ans. 1 of an acre. 14. A ferryboat is valued at 100 dollars. What share or interest in the boat could be purchased for 37 dollars?

Ans. of it. 15. A person who undertook a journey of 425 miles, having traveled 89 miles; what part of the journey has he accomplished? Ans. of it. 16. A manufactory is estimated to be worth 12000 dollars. What interest in it could be purchased for 2143 dollars? Ans. 13 of it.

12000