49. When the multiplier contains but one figure. Write the multiplier under the multiplicand; then, beginning at the right hand, multiply each figure of the multiplicand by the multiplier separately. If the product of any figure of the multiplicand into the multiplier does not exceed 9, set it in its proper place under the figure multiplied; but if it does exceed 9, write the units' figure under the figure multiplied, and carry the tens to the next product on the left, as in addition. (Art. 25.) 50. The principle of carrying the tens in multiplication is the same as in addition, and may be illustrated in a similar manner. (Art. 26.) Take, for instance, the last example, and set the product of each figure in a separate line. Adding these results, we have 4698 Product. In this analytic solution it will be seen that the tens' figure in each product which exceeds 9, is added to the next product on the left, the same as in the common method of solving this and similar examples. The only difference between the two operations is, that in one case we add the tens as we proceed in the multiplication; in the other we reserve them till each figure is multiplied, and then add them to the same orders as before: consequently the result must be the same in both. (Art. 27.) QUEST.-49. How do you write the numbers for multiplication? Where begin to multiply? When the product of a figure in the mul tiplicand does not exceed 9, where is it written? When it exceeds 9, what is to be done with it? 50. How illustrate the principle of carrying in multiplication? 51. From this and the preceding illustrations, the learner will perceive, that units multiplied by units produce units; tens into units produce tens; hundreds into units produce hundreds, &c. Hence, When the multiplier is units, the product will always be of the same order as the figure multiplied. 12. What cost 83 pounds of opium, at 8 dollars per pound? 13. At 9 shillings per day, how much can a man earn in 213 days? 14. If 1 sofa costs 78 dollars, how much will 8 sofas cost? 15. What cost 879 barrels of flour, at 7 dollars a barrel? 16. At 8 shillings apiece, what will a drove of 650 lambs come to? 21. What will 26 horses cost, at 113 dollars apiece? Suggestion.-Reasoning as before, if 1 horse costs 113 dollars, 26 horses will cost 26 times as much. Since it is not conven Operation. 113 Multiplicand. 678 cost of 6 horses. 226* cost of 20 66 Ans. 2938 cost of 26 66 ient to multiply by 26 at once, we first multiply by the 6 units, then by the 2 tens and add the two results together.Thus 6 times 3 are 18; set down the 8 and carry the 1, as above. 6 times 1 are 6, and 1 to carry makes 7. 6 times 1 are 6. Next, multiply by the 2 tens thus: 20 times 3 units are 60 units or 6 tens; or we may simply say, 2 times 3 are 6. Now the 6 must denote tens; for units into tens, or what is QUEST.-51. What do units multiplied into units produce? Tens into units? Of what order is the product universally, when the multiplier is units? the same thing, (Art. 47,) tens into units, produces tens: consequently the 6 must be written in tens' place in the product; that is, under the figure 2 by which we are multiplying. 20 times 1 ten are 20 tens or 200; or simply say, 2 times 1 are 2: and since the 2 denotes hundreds, as we have just seen, set it on the left of the 6 in hundreds' place. 20 times 1 hundred are 20 hundred or 2000; or simply say, 2 times 1 are 2: and since the 2 denotes thousands, set it in the thousands' place on the left of the last figure in the product. Finally, adding these two results together as they stand, units to units, tens to tens, &c., we have 2938 dollars, which is the whole product required. Note.-The several products of the multiplicand into the separate figures of the multiplier, are called partial products. Hence, 52. When the multiplier contains more than one figure. Multiply each figure of the multiplicand by each figure of the multiplier separately, and write each partial product in a separate line, placing the first figure of each line directly under that by which you multiply; finally, add the several partial products together, and the sum will be the true product or answer required. 53. PROOF-Multiply the multiplier by the multiplicand, and if the product thus obtained is the same as the other product, the work is supposed to be right. OBS. 1. This method of proof depends upon the principle, that the product of any two numbers is the same, whichever is taken for the multiplier. (Art. 47.) 2. When the multiplier is small, we may add the multiplicand to itself as many times as there are units in the multiplier, and if the sum is equal to the product, the work is right. Thus 78X3=234. Proof-78+78+78-234, which is the same as the product. 3. Multiplication may also be proved by division, and by casting out the nines; but neither of these methods can be explained here QUEST.-Note. What is meant by partial products? 52. How do you proceed when the multiplier contains more than one figure? How should the partial products be written? Where write the first figure of each line? What do you finally do with the partial products? 53. How is multiplication proved? Obs. On what principle does this method of proof depend? When the multiplier is small, how may we prove it? without anticipating principles belonging to division, with which the learner is supposed as yet to be unacquainted. 22. What will 45 cows cost, at 27 dollars a head? 23. What cost 63 hats, at 36 shillings apiece? 24, How much corn can a man raise on 87 acres, at 45 bushels per acre? 25. How many pounds of sugar will 75 boxes contain, if each box holds 256 pounds? 26. What cost 278 hogsheads of molasses, at 23 dollars per hogshead? 27. What is the product of 347 multiplied by 256? Suggestion.-Proceed in the same manner as when the multiplier contains but two figures, remembering to place the right hand figure of each partial product directly under the figure by which you multiply. Operation. 347 256 2082 1735 694 88832 Ans. 28. What is the product of 569 into 308 ? After multiplying by the 8 units, we must next multiply by the 3 hundreds, since there are no tens in the multiplier, and place the first figure of this partial product directly under the figure 3 by which we are multiplying. Operation. 569 308 4552 1707 175252 Ans. 29. What is the product of 67025 into 4005? Ans. 268435125. 30. What is the product of 841072 into 603? 54. From the preceding illustrations and principles we derive the following GENERAL RULE FOR MULTIPLICATION. I. Write the multiplier under the multiplicand, units under units, tens under tens, &c. II. When the multiplier contains but one figure. Begin with the units, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (Art. 49.) III. When the multiplier contains more than one figure. Multiply each figure of the multiplicand by each figure of the multiplier separately, beginning at the right hand, and write the partial products in separate lines, placing the first figure of each line directly under the figure by which you multiply. (Art. 52.) Finally, add the several partial products together, and the sum will be the whole product. OBS. It is immaterial as to the result which of the factors is taken for the multiplier. (Art. 47.) But it is more convenient and therefore customary to place the larger for the multiplicand and the smaller for the multiplier. Thus, it is easier to multiply 254672381 by 7, than it is to multiply 7 by 254672381, but the product will be the same. EXAMPLES FOR PRACTICE. 1. What will 465 hats cost, at 6 dollars apiece? 2. What will 638 sheep cost, at 4 dollars a head? 3. What will 1360 yards of cloth cost, at 7 dollars a yard? 4. What cost 169 bushels of potatoes, at 4 shillings per bushel ? 5. What cost 279 barrels of salt, at 9 shillings a barrel? 6. At 12 dollars a suit, how much will it cost to furnish 1161 soldiers with a suit of clothes apiece? 7. What cost 1565 acres of wild land, at 7 dollars per acre? QUEST.-54. What is the general rule for multiplication? Obs. Which number is usually taken for the multiplicand? |