33. What will be the cost of 9 dressing gowns at 5 dollars apiece, 3 pairs slippers at 1 dollar a pair, 2 pairs boots at 4 dollars a pair, and 3 dozen stockings at 2 dollars a dozen?

34. Suppose in 1 yard of cloth there are 580 fibres of warp and 432 of filling, and that each fibre of warp contains 32 strands, and each of filling 48, how many strands in the yard?

35. The Lawrence Pacific Mills turn out material for about 65000 dresses in a week; how many will they make in a year, or 52 weeks?

36. Allowing 12 yards to a dress, how many yards do they make in a year r?

For Contractions in Multiplication, see Appendix.

For Dictation Exercises, see Key.

DIVISION.

55. Division is the process of ascertaining how many times one number is contained in another, or of finding one of the equal parts of a number.

NOTE. In the example, "John has 10 apples, which he wishes to give to as many boys as he can, giving them 2 apples apiece, to how many can he give them?" - it is evident he can give them to as many boys as 2 is contained times in 10. In the example, "If 16 pears are divided equally among 4 boys, how many pears does 1 boy receive?" it is evident that 1 boy must receive one fourth of what the 4 boys receive, or one fourth of 16 pears; that is, one of the four equal parts of the number, 16 pears.

The number which is divided is called the Dividend, the num ber by which we divide is called the Divisor, and the result the Quotient, from the Latin quoties, how many times.

The sign of Division is a short horizontal line between two dots,÷; thus, 9 ÷ 3 shows that 9 is to be divided by 3. Sometimes the dividend and divisor take the place of the dots; thus, This expression may be read, 9 divided by 3, nine thirds, or one third of nine, and is the fractional * form of division.

*See Art: 82.

NOTE.

SHORT DIVISION.

This method is to be preferred where the divisor is not greater than 12.

56. ILLUSTRATIVE EXAMPLE, I.

Divide 936 by 6.

OPERATION.

Divisor 6) 936 Dividend.
Quotient 156

We place the divisor at the left of the dividend, from which we separate it by a curved line, and, drawing a straight line beneath the dividend, proceed thus: 6 is contained in 9 hundreds 1 hundred times, with 3 hundreds remaining. We write the 1 hundred beneath the hundreds in the dividend, and reduce the 3 hundreds remaining to tens. 3 hundreds equal 30 tens, which, with the 3 tens of the dividend, equal 33 tens. 6 in 33 tens, 5 tens times, with a remainder of 3 tens; writing the 3 tens in the tens' place, and reducing the remainder as before, we have 36 units. 6 in 36, 6 times; writing the 6 in the units' place, we have 156 as the quotient of 936 divided by 6.

ILLUSTRATIVE EXAMPLE, II.

Divide 17869 by 7.

OPERATION.

7) 17869

Ans. 2552-5 Remainder.

In this example, as 7 is not contained in 1 (ten thousand) any number of (ten thousand) times, we shall have no ten thousands in the quotient, and therefore take 17 (thousands) for our first partial dividend. We find also that the dividend does not contain the divisor an exact number of times, but that there is a remainder of 5. As this does not contain 7 any whole number of times, we can indicate the division by placing the 5 in the quotient above the divisor, and have for the answer 2552§, which is read, two thousand five hundred fifty-two and five sevenths.

From the above examples we derive the

RULE FOR SHORT DIVISION. Beginning at the left, divide the first term or terms of the dividend by the divisor, make the result the first term of the quotient.

Prefix the remainder, should there be any, to the next term of the dividend, divide as before, and thus continue till all the terms of the dividend are divided.

Should there be a remainder after the last division, place the divisor beneath it, and annex the result to the quotient.

57. PROOF I. the divisor and quotient being factors of the dividend: hence, to prove an example in division, multiply the quotient by the divisor, and to the product add the remainder. The sum thus obtained

Division is the converse of Multiplication,

14. How many barrels of flour, at 7 dollars a barrel, can I

buy for 259 dollars?

15. At 11 cents a yard, how many yards of cloth can I buy for 368972 cents?

16. If 12 pieces of cloth contain 408 yards, how many yards in a piece?

17. How many weeks are there in 4781 days?

18. How many hours will it take me to walk 1378 miles, at 5 miles an hour?

19. 9 times a certain number equals 324783; what is that number?

20. 8 X what 36924?

Ans. 36,087.

21. 12 X what 46817?

LONG DIVISION.

59. Long Division is the process of dividing where the divisor is large, and the work written down.

60. ILLUSTRATIVE EXAMPLE, I.

Divide 85232 by 23.

OPERATION.

69

162

161

132

115

17

23 is contained in 85 (thous.) 3 (thous.) times;

23)85232(37051. we place the 3 (thous.) in the quotient, at the right of the dividend. 3 (thous.) X 23 = 69 (thous.), which, subtracted from 85 (thous.), leaves a remainder of 16 (thous.). Bringing down the next figure of the dividend, we have 162 (hund's), which contains 23, 7 (hund's) times; we place the 7 (hund's) in the quotient at the right of the 3 (thous.). 7 (hund's) X 23-161 (hund's), which, subtracted from 162 (hund's), leaves 1 (hund.). Bringing down the 3 (tens) of the dividend, we have 13 (tens), which does not contain 23 any number of (tens) times. Placing a zero, therefore, in the ten's place in the quotient, we bring down the next figure, 2, and have 132 units; 23 in 132, 5 times. Writing the 5 in the unit's place in the quotient, multiplying and subtracting as before, we have a remainder of 17, and for our answer, 37051}. Hence the RULE FOR LONG DIVISION. Beginning at the left, divide the first term or terms of the dividend by the divisor; make the result the first term of the quotient. Multiply the divisor by this term, and subtract the product from that part of the dividend used.

Annex the next term of the dividend to the remainder; divide as before, and thus continue till all the terms of the dividend are divided.

Should there be a remainder after the last division, place the divisor beneath it, and annex the result to the quotient.

NOTE 1. When it is difficult to determine the quotient figure at sight, trial divisors may be used.

For example, divide 29847 by 476.

It is evident that 476 is contained in the dividend 476) 29847 ( fewer times than 400 is contained in it, and more times than 500. Rejecting two right hand figures from the divisor, also from that part of the dividend first considered, we see that 4 is contained in 29, 7 times, and 5 in 29, 5 times; therefore the quotient figure cannot be more than 7 nor less than 5.

NOTE 2. If, at any time, the product obtained by multiplying the divisor by any term of the quotient, exceeds the partial dividend, the quotient figure is too large. If, at any time, the remainder equals or exceeds the divisor, the quotient figure is too small.

61. PROOF II. By casting out the 9's. Multiply the excess of 9's in the divisor, by the

(Art. 50, note.)

excess of 9's in

the quotient, and find the excess of 9's in the product; if it equals the excess of 9's in the dividend after the remainder has been subtracted, the work is presumed to be right.

62. ILLUSTRATIVE EXAMPLE, II.

Divide 26874 by 44.

6. Divide 1459998 by 38, 19, 57, 171, 49, 513, 76842, and add the quotients.

Ans. 182,075.

7. Divide 195989184 by 41, 16, 144, 164, 123, 369, 72, 656, 1968, and add the quotients.

Ans. 24,830,608.

8. Divide 43586118576 by 17, 56, 119, 136, 4158, 51, 72, 126, 99, 45738, 29106, 320166, and add the quotients.

9. 975318642 ÷ 893 = ? 10. 800000231 73 =?

Ans. 6,288,215,934

12. 83473009004 = ? 13. 769800281 876=?