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10 times 0 are 0 11 times 0 are 0/12 times 0 are 0 10 times l are 10 11 times 1 are 11 12 times l are 12 10 times 2 are 2011 times 2 are 22 12 times 2 are 24 10 times 3 are 30 11 times 3 are 33 12 times 3 are 36 10 times 4 are 40 11 times 4 are 44 12 times 4 are 48 10 times 5 are 50 11 times 5 are 55 12 times 5 are 60 10 times 6 are 60 11 times 6 are 66 12 times 6 are 72 10 times 7 are 70 11 times 7 are 77 12 times 7 are 84 10 times 8 are 80 11 times 8 are 88 12 times 8 are 96 10 times 9 are 90 11 times 9 are 99 12 times 9 are 108 10 times 10 are 100 11 times 10 are 110 12 times 10 are 120 10 times 11 are 110 11 times 11 are 121 12 times 11 are 132 10 times 12 are 120 11 times 12 are 132 12 times 12 are 144

EXAMPLES. 1. Let it be required to multiply 4 by 2. Here 4 is the multiplicand and 2 is the multiplier, and it is required to find the product, which is the number arising from repeating 4 two times.

OPERATION.

The product of 4 by 2 is found by multiplication or by adding two 4's together.

W Multiplicand.

Multiplier.

4 4

Product.

2. Let it be required to multiply 4 by 3, and also to multiply 5 by 3.

OPERATION.

OPERATION.

4
4
4
12 Product.

5 5 5 15 Product.

From these examples we see, that the product of 4 multiplied by 3 is 12, the number which arises from adding three 4's together; and that the product of 5 by 3 is equal to 15, the number which arises from adding three 5's together.

We see from the above examples, that any product may be found by setting down the multiplicand as many times as there are units in the multiplier, and adding all the numbers together. MULTIPLICATION is therefore a short method of addition.

Q. How may any product be found? What may multiplication be considered ? $ 22. In the example in which

OPERATION. 5 was multiplied by 3, the pro

5. duct was 15. Now, had we multiplied 3 by 5, the product would

1 1 1 1 still have been 15. For, place as 3 1 1 1 1 1 many ones in each horizontal row

1 1 1 1 1 as there are units in the multiplicand, and make as many rows as there are units in the multiplier : the product will then be equal to the whole number of ones : viz., 15. But if we consider the number of ones (3) in a vertical row to be the multiplicand, and the number of vertical rows (5) the multiplier, the product will still be the whole number of ones : viz., 15.

Hence,

Either of the factors may be used as the multiplier without altering the product. For example,

3x7=7X3=21: also, 6x3=3x6=18.

9X5=5x9=45: also, 8X6=6x8=48. and, 8x7=7X8=56: also, 5x7=7X5=35.

Q. Is the product of two numbers altered by changing the multiplicand into the multiplier, and the multiplier into the multiplicand ? . Is 7 multiplied by 8 the same as 8 multiplied by 7?

3. Multiply 236 by 4.

First set down the 236, then place OPERATION. the 4 under the unit's place 6, and draw 236 a line beneath it. Then multiply the 4 6 by 4: the product is 24 units; set

24 units. them down. Next multiply the 3 tens 12 by 4: the product is 12 tens; set down

8 hundreds. the 2 under the tens of the 24, leaving

944 the 1 to the left, which is the place of

tens.

OPERATION.

hundreds. Next multiply the 2 by 4: the product is 8, which being hundreds, is set down under the l. The sum of these numbers, 944, is the entire product.

The product can also be found, thus : say 4 times 6 are 24 : set down the 4, and

236 then say, 4 times 3 are 12 and 2 to carry

4 are 14: set down the 4, and then say, 4

944 times 2 are 8 and 1 to carry are 9. Set down the 9, and the product is 944 as before.

4. Multiply 627 by 84.

Multiply by the 4 units, as in the last OPERATION, example. Then multiply by the 8 tens. 627 The first product 56, is 56 tens; the 6, 84 therefore, must be set down under the 0,

2508 which is the place of tens, and the 5 car 5016 ried to the product of the 2 by 8. Then

52668 multiply the 6 by 8, carrying the 2 from the last product, and set down the result 50. The sum of the numbers 52668, is the required product.

5. Multiply 506 by 302.

In this example, we say, 2 times 6 are OPERATION. 12 : then set down the 2, and say, 2 times 506 0 are 0 and 1 to carry make 1. Set down

302 the 1, and say, 2 times 5 are 10: set down 1012 the 10. Then beginning with the 0, we 000 say, 0 times 6 is 0: set down the 0. Then

1518 say, 0 times 0 is 0; set down the 0, and

152812 then say, 0 times 5 is 0. Then multiply by the 3 hundreds and set down the first figure 8 in the place of hundreds, and place the other figures to the left.

When an 0 appears in the multiplier, we need not multiply by it, since each of the products is 0; but when we multiply by the next figure to the left, we must observe to set the first figure of the product directly under its multiplier.

Thus, we have placed 8 directly under the multiplier 3.

Q. When an 0 is found in the multiplier need you multiply by it? When you multiply by the next figure to the left, where do you place the first figure of the product ?

CASE I.

§ 23. When the multiplier does not exceed 12.

RULE.

I. Set down the multiplicand and under it set the multiplier, so that units shall fall under units, and draw a line beneath.

II. Multiply every figure of the multiplicand by the multiplier, setting down and carrying as in addition.

[blocks in formation]

Q. When the multiplier does not exceed 12, how do you set it down? How do you multiply by it?

CASE II.

§ 24. When the multiplier exceeds 12.

RULE. I. Set down the multiplier under the multiplicand, so that units shall fall under units, tens under tens, &c., and draw a line beneath.

II. Begin with the right hand figure, and multiply all the figures of the multiplicand by each figure of the multiplier, and when any of the products exceeds 9, set down and carry to the next product as in addition ; observing to write the first figure of each product directly under its multiplier.

III. Add up the several products and their sum will be the product sought.

Note. There are three numbers in every multiplication. First, the multiplicand : second, the multiplier: and third, the product.

PROOF OF MULTIPLICATION. Write the multiplicand in the place of the multiplier, and find the product as before : if the two products are the same, the work is supposed right.

Q. When the multiplier exceeds 12, how do you set it down ? How do you multiply by it? How do you add up? How many numbers are there in every multiplication ? Name them? How do you prove multiplication ?

EXAMPLES.

1. Multiply 365 by 84: also, 37864 by 209. (1.) (2.)

(3.) Multiplicand 365 37864 34293 47042 Multiplier 84

209
74

91 1460

2920 Product. 30660

4280822 (5.) (6.)

(8.) 46834 679084 1098731

8971432 406 126 1987

10471 19014604

93939864472 9. Multiply 12345678 by 32. Ans. 395061696. 10. Multiply 9378964 by 42. Ans. 11. Multiply 1345894 by 49. Ans. 65948806. 12. Multiply 576784 by 64.

Ans. 13. Multiply 596875 by 144. Ans. 85950000. 14. Multiply 46123101 by 72. Ans. 15. Multiply 61835720 by 132. Ans. 8162315040. 16. Multiply 718328 by 96.

Ans. 17. Multiply 7128368 by 1440. Ans. 10264849920. 18. Multiply 6795634 by 918546.

Ans. 6242102428164. 19. Multiply 86972 by 1208. Ans. 20. Multiply 1055054 by 570. Ans. 601380780. 21. Multiply 538362 by 9258. Ans. 22. Multiply 50406 by 8050. Ans. 405768300. 23. Multiply 523972 by 15276. Ans. 24. Multiply 760184 by 16150. Ans. 12276971600. 25. Multiply 1055070 by 31456. Ans. 33188281920.

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