1 2 3 4 4 6 6 9 8 12 5 10 15 6 12 18 14 21 24 9 18 27 10 20 30 11 22 33 12 Units, Tens, 2 3 681 16 5 10 12 15 16 20 20 25 24 30 28 35 32 40 36 45 40 50 44 55 24 36 48 60 Multiplicand, 3 Multiplier, MULTIPLICATION TABLE. Co Hunds. Hundreds, 18 Product, OPERATION. 374 6 2244 +8 Tens. 6 24 42 EXAMPLES. 60. When the multiplier consists of one figure. 1. Multiply 374 by 6. OPERA TION. SOLUTION. - In this example it is required to take 374 six times. If we take the units of each order 6 times, we shall take the entire number 6 times. Therefore, writing the multiplier under the unit figure of the multiplicand, we proceed as follows: 6 times 4 units are 24 units; 6 times 7 tens are 42 tens; 6 times 3 hundreds are 18 hundreds ; and adding these partial products, we obtain 2244 the entire product, 2244. 40 48 56 64 72 80 80 66 77 88 72 45 54 63 72 81 90 90 99 110 121 132 84 96 108 120 132 144 in another The operation in this example may be performed way, which is the one in common use. SOLUTION. -Writing the numbers as before, we begin at the right hand or unit figure, and say: 6 times 4 units are 24 units, which are 2 tens and 4 units; we write the 4 units in the product in units' place, and reserve the 2 tens to add to the next product; 6 times 7 tens are 42 tens, and the two tens reserved in the last product added, are 44 tens, which are 4 hundreds and 4 tens; we write the 4 tens in the product in tens' place, and reserve the 4 hundreds to add to the next product; 6 times 3 hundreds are 18 hundreds, and 4 hundreds added are 22 hundreds, which being written in the product in the places of hundreds and thousands, gives, for the entire product, 2244. 61. The unit value of a number is not changed by repeating the number. 62. When both the factors of a product are abstract, the product will be abstract. When one of the factors is concrete, the product will be of the same unit value as that factor. 1. When, as explained in the note under 59, a concrete number is used as a multiplier, the product will be of the same denomination as the multiplier. When the multiplicand is the concrete number, the product will be of the same denom. ination as the multiplicand. 2. In multiplying, learn to pronounce the partial results, as in addition, without naming the numbers separately; thus, in the last example, instead of saying 6 times 4 are 24, 6 times 7 are 42 and 2 to carry are 44, 6 times 3 are 18 and 4 to carry are 22, pronounce only the results, 24, 44, 22, performing the operations mentally. This will greatly facilitate the process of multiplying. 9. Multiply 32746 by 5. 10. Multiply 840371 by 7. 11. Multiply 137629 by 8. 12. Multiply 93762 by 3. 13. Multiply 543272 by 4. 14. Multiply 703164 by 9. Ans. 281286. Ans. 2173088. 15. What will be the cost of 344 cords of wood, at $4 a cord? Ans. $1376. Ans. 163730. Ans. 5882597. Ans. 1101032. 16. How much will an army of 7856 men receive in one week, if each man receives $6? Ans. $47136. How many 17. In one day there are 86400 seconds. seconds are there in 7 days? Ans. 604800 seconds. 18. What will 7640 pounds of prunes cost, at 9 cents a pound? Ans. $687.60. 19. At $5 an acre, what will 2487 acres of land cost? Ans. $12435. 20. In one mile there are 5280 feet. How many feet are there in 8 miles? Ans. 42240 feet. 21. What will 125 barrels of apples cost, at $4 a barrel ? 22. There are 24 hours in a day. How many hours are there in 365 days? 63. When the multiplier consists of two or more figures. 1. Multiply 746 by 23. Multiplicand, Multiplier, OPERATION. 746 23 SOLUTION. Writing the multiplicand and multiplier as in 60, we first multiply each figure in the multiplicand by the unit figure of the multiplier, precisely as in 60. Then we multiply by the 2 tens. 2 tens times 6 units, or 6 times 2 times the mul Product, 17158 23tiplicand. tens, are 12 tens, equal to 1 hundred, and 2 tens; we place the 2 tens under the tens' figure in the product already obtained, and add the 1 hundred to the next hundreds produced. 2 tens times 4 tens are 8 hundreds, and the 1 hundred of the last product added are 9 hundreds; we write the 9 in the hundreds' place in the product. 2 tens times 7 hundreds are 14 thousands, equal to 1 ten thousand and 4 thousands, which we write in their appropriate places in the product. Then adding the two products, the entire product is 17158. When there are ciphers between the significant figures of the multiplier, pass over them, and multiply by the significant figures only. 2238 Stimes the mul 3 {tiplicand. S times the mul- 1492 20 64. When the multiplier contains two or more figures, the several results obtained by multiplying by each figure are called partial products. From the preceding examples and illustrations we deduce the following general rule: RULE.-I. Write the multiplier under the multiplicand, placing units of the same order under each other. II. Multiply the multiplicand by each figure of the multiplier successively, beginning with the unit figure, and write the first figure of each partial product under the figure of the multiplier used, writing down and carrying as in addition. III. If there are partial products, add them, and their sum will be the product required. PROOF.-I. Multiply the multiplier by the multiplicand, and if the product is the same as the first result, the work is probably correct. Or, II. Multiply the multiplicand by the multiplier dimin ished by 1, and to the product add the multiplicand; if the sum is the same as the product by the whole of the multiplier, the work is probably correct. 8. How many yards of linen are there in 759 pieces, each piece containing 25 yards? Ans. 18975 yards. 9. Sound is known to travel about 1142 feet in a second of time. How far will it travel in 69 seconds? 10. A man bought 36 lots at $475 each. What did they all cost him? Ans. $17100. 11. What would be the value of 867 shares of railroad stock, at $97 dollars a share? Ans. $84099. 12. How many pages are there in 3475 books, if there are 362 pages in each book? Ans. 1257950 pages. 13. In a garrison of 4507 men, each man receives annually $208. What do they all receive? 14. Multiply 7198 by 216. 15. Multiply 31416 by 175. 16. Multiply 7071 by 556. 17. Multiply 75649 by 579. 18. Multiply 15607 by 3094. 19. Multiply 79094451 by 76095. Ans. 6018692248845. 20. Multiply five hundred forty thousand six hundred. nine, by seventeen hundred fifty. Ans. 946065750. 21. Multiply four million twenty-five thousand three hundred ten, by seventy-five thousand forty-six. Ans. 1554768. Ans. 43800771. Ans. 302083414260. 22. Multiply eight hundred seventy-seven million five hundred ten thousand eight hundred sixty-four, by five hundred forty-five thousand three hundred fifty-seven. Ans. 478556692258448. 23. If one mile of railroad requires 116 tons of iron, worth $65 a ton, what will be the cost of sufficient iron to construct a road 128 miles in length? Ans. $965120. 24. The area of the United States is 3,668,167 square miles, and according to the census of 1890 there was an average of 17 people to the square mile. What was the population of the United States according to that census? |