90 11 times 9 are 99 12 times 9 are 108 10 times 9 are 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. 2. Let it be required to multiply 4 by 3, and also to multiply 5 by 3. 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 5 was multiplied by 3, the product was 15. Now, had we multiplied 3 by 5, the product would still have been 15. For, place as many ones in each horizontal row as there are units in the multipli OPERATION. 5. 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 cand, 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, 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 the 4 under the unit's place 6, and draw a line beneath it. Then multiply the 6 by 4: the product is 24 units; set them down. Next multiply the 3 tens by 4: the product is 12 tens; set down the 2 under the tens of the 24, leaving the 1 to the left, which is the place of hundreds. Next multiply the 2 by 4: the product is 8, which being hundreds, is set down under the 1. 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 then say, 4 times 3 are 12 and 2 to carry are 14 set down the 4, and then say, 4 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 example. Then multiply by the 8 tens. The first product 56, is 56 tens; the 6, therefore, must be set down under the 0, which is the place of tens, and the 5 carried to the product of the 2 by 8. Then multiply the 6 by 8, carrying the 2 from the last product, and set down the result 50. the numbers 52668, is the required product. 5. Multiply 506 by 302. OPERATION. 236 4 944 OPERATION, 627 84 2508 5016 The sum of OPERATION. 506 302 1012 000 In this example, we say, 2 times 6 are 12: then set down the 2, and say, 2 times 0 are 0 and 1 to carry make 1. Set down the 1, and say, 2 times 5 are 10: set down the 10. Then beginning with the 0, we say, 0 times 6 is 0: set down the 0. Then say, 0 times 0 is 0; set down the 0, and 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. 1518 152812 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 mul tiplier, setting down and carrying as in addition. 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. (3.) 34293 (4.) 47042 Multiplicand 365 |