EXAMPLES.

51. Perform and prove the following examples:3. 18762 X 236 =? Ans. 132,624. 4. 128124 × 402 = ?

1. 3684 X 36 =?

2. 2842 X 28=?

5. 189003 X 836 — ?

Ans. 79,576. 6. 12053 X 972 =?

7. Add the answers to the last four examples, and multiply the sum by 3798. Ans. 857,040,362,792.

8. Multiply 123456789 by 98765.

52. Any number may be multiplied by 10, 100, 1000, or a unit of any order, by annexing as many zeros to the multiplicand as there are zeros in the multiplier, and placing the decimal point at the right.

EXAMPLES.

9. Multiply 68432 by 10, by 100, 10000, 1000, 1000000, and add the products. Ans. 69,192,279,520. 10. Multiply 3682 by 10000, 10, 1000, 100, 100000, and add the products.

53. ILLUSTRATIVE EXAMPLE, III.

Multiply 68432 by 86000.

OPERATION.

68432

Here, by multiplying first by 86, and then annex 86000 ing three zeros, which multiplies the first product by

410592

547456

one thousand, the true result is obtained, and labor saved.

5885152000 Ans.

ILLUSTRATIVE EXAMPLE, IV.

Multiply 832000 by 210.

OPERATION.

832000

210

832

1664

Here the zeros in both the multiplicand and multiplier are disregarded until after muliplying the other terms together.

174720000 Ans.

21. Add the last ten answers, and multiply the sum by 100. Ans. 81,482,871,584,800. 22. How many hills of corn have I in my cornfield, which contains 97 rows and 45 hills in a row?

23. If each hill produces 18 ears, how many ears does the field produce?

24. I have four corn bins, containing severally 63 bushels, 54 bushels, 37 bushels, and 29 bushels. There are four pecks in a bushel. How many pecks do they all hold?

25. Allowing 23 ears of corn to a peck, how many ears are there in the bins?

26. If a barrel of flour costs 9 dollars, what will 368 barrels cost?

27. If a person by working 12 hours a day can do a piece of work in 37 days, in how many days can he do it working 1 hour a day?

28. I have 5 bins, which contain 69 bushels each. What will be the capacity of a bin which will contain as much as all of them? 29. If 6 yards of cloth will make one pair of shirts, how many yards will make one dozen or 12 shirts? How many will make

8 dozen?

30. What will 3 dozen cost at 15 cents per yard for the cloth, 30 cents apiece for bosoms, wristbands, and buttons, and 50 cents apiece for making?

31. It takes 7 yards of ticking for a single bed-sack; what must I pay for cloth for 18 single bed-sacks, at 16 cents per yard?

32. If sheeting can be bought for 17 cents a yard, whar must I pay for cloth for 21 sheets, allowing 10 yards for a pair?

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?

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, 93 shows that 9 is to be divided by 3. Sometimes the dividend and divisor take the place of the lots; thus, This expression may be read, 9 divided by 3, nine thirds, or ne 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 twelve.

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

In this example, as 7 is not contained in 1 (ten thousand) any number of (ten thousand) times, we shall have no ten Ans. 2552-5 Remainder. 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 25524, which is read, two thousand five hundred fifty-two and five sevenths.

From the above example 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.

Division is the converse of Multiplication,

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

should equal the dividend.

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? Ans. 36,087.

20. 8 X what = 36924?

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.