45. Multiplication is the process of finding a number equal in value to one number taken as many times as there are units in another number. The number which is multiplied is called the Multiplicand, the number by which we multiply is called the Multiplier, and the result obtained is called the Product.

The multiplicand and multiplier are often called factors of the product, from the Latin facio, I make, because, being multiplied together, they make up the product. The product is also said to be the multiple of the factors. Thus, 7 times 6 42. Here, 7 is the multiplier, 6 the multiplicand, and 42 the product; or 7 and 6 are the factors of 42, which is their multiple.

The sign of multiplication is a small, oblique cross, X, read, times, or, multiplied by. Thus, 7 X 6 may be read either 7 times 6, or 7 multiplied by 6. In the former case 7 is the multiplier and 6 the multiplicand, while in the latter 6 is the multiplier and 7 the multiplicand. The product is the same, whichever is the multiplier.

NOTE.-In the process of multiplication, the multiplier must be an abstract number. We cannot multiply pencils by pencils, or pencils by apples, but either may be multiplied by an abstract number, and give a product of the same denomination as the concrete factor. (Art. 4.)

46. ILLUSTRATIVE EXAMPLE, I.

Multiply 2364 by 7.

Factors

{

OPERATION.

2364 Multiplicand.
7 Multiplier.

Multiple, 16548 Product.

2 tens and 8 units. We write the 8 in the units' place, and reserve the 2 tens for the tens' place. 7 times 6 tens 42 tens, which, with the 2 reserved tens, 44 tens: =4 hundreds and 4 tens; we write the 4 tens in the tens' place, and reserve the 4 hundreds for the hundreds' place. 7 times 3 hundreds =21

=

hundreds, which, with the 4 reserved hundreds, 25 hundreds.=2 thousands +5 hundreds; we write the 5 hundreds in the hundreds' place, and reserve the 2 thousands for the thousands' place. 7 times 2 thousands 14 thousands, which, with the 2 thousands reserved,: = 16 thousands 1 ten-thousand +6 thousands; we write the 6 thousands in the thousands' place, and the 1 ten-thousand in the ten-thousands' place, and thus obtain for our product 16548.

NOTE. This result might be obtained by finding the sum of the number 2364 taken seven times; that is, by adding 2364 to itself six times. Hence, Multiplication may be regarded as a short way of performing Addition.

47. EXAMPLES.
Multiply

1. 267 by 2; by 3; by 4; and add the products. Ans. 2,403.

2. 628 by 5; by 6; by 7; 3. 3401 by 8; by 9;

4. 90021 by 10; by 11 5. 66285 by 12; by 8;

Ans. 11,304.

Ans. 57,817.

Ans. 1,890,441.

9832 X 7 what?

22078 X 11=what?

19869 X 12 = what?

14. Add the last eight products, and multiply by 7.

Ans. 5,205,081.

15. 123456 X 6 = ?

16.

987654321 X 7 = ?

17. Add the last two products.

Ans. 6,914,320,983

48. ILLUSTRATIVE EXAMPLE, II.

Multiply 3648 by 294.

OPERATION.

3648

294

14592

Here we are to multiply, not only by units, but
by tens and hundreds. We write the numbers
units under units, tens under tens, &c., and mul-
tiply first by the units, as before, and then by the
tens. It is evident that the product of any num-
ber multiplied by tens will be ten times as great
as if multiplied by the same number of units;
1072512 Ans.
multiplied by hundreds, one hundred times as
great as if multiplied by units; multiplied by

32832
7296

thousands, one thousand times as great, etc. Hence, when a number is multiplied by tens, hundreds, or thousands, the products thus ob tained are written one, two, or three places farther to the left than when multiplied by units; or, in other words, we multiply by the other terms as we multiply by the units, placing the first figure of each product under the term by which we multiply. The sum of these partial products is the entire product. Hence the

RULE FOR MULTIPLICATION.

Write the multiplier under the multiplicand. Beginning at the right, multiply each term of the multiplicand by each term of the multiplier, successively, placing the right hand figure of each partial product under the term by which you multiply, carrying as in addition. Add all the partial products, and the result will be the entire product.

49. PROOF I. Take the multiplicand for the multiplier, and the multiplier for the multiplicand. If the result thus obtained be like the first result, the work is probably correct.

50. PROOF II. By casting out the 9's. This method is much the easier, though not always sure.

NOTE. To cast out the 9's from any number, commence at the left, and add the digits towards the right. When their sum equals 9 or more, reject 9 and add the remainder to the next digit, and so on. The last remainder is called the excess of 9's.

TO PROVE MULTIPLICATION BY CASTING OUT THE 9'S.

Cast out the 9's from each of the factors. Then multiply the remainders, should there be any, cast out the 9's from the product, and note the last remainder. Cast out the 9's from the answer, and if the remainder equals the one obtained above, the work may be presumed to be right; thus,

36184

3 +6 = 9. 1+8=9. 4, 1st remainder. 2+6+8=16=9+7. 7+1= 8, 2d remainder.

2681

36184

289472

217104

72368

97009304 Ans.

32

3+25, last remainder.

7+3=10=9+1. 1+4= 5, which equal

ling the remainder above, the work is right.

NOTE. For demonstration of rule, see Appendix.

EXAMPLES.

51. Perform and prove the following examples: —— 3. 18762 X 236 = ?

1. 3684 X 36=?

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

410592

547456

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 cars 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, what must I pay for cloth for 21 sheets, allowing 10 yards for a pair?