23. Multiply 5872 by 10; by 1000; and add the products. 24. Multiply 684 by 10; by 100; by 10000; by 1000; and add the products. 25. Multiply 3682 by 10000; by 10; by 1000; by 100; and add the products. 80. ILLUSTRATIVE EXAMPLE IV. WRITTEN WORK. 4520 300 1356000 Explanation. Multiply 4520 by 300. Here 4520 equals 452 × 10, and 300 equals 3 × 100; hence 4520 × 300 is the same as 452 × 10 × 3 × 100, or, since the order of the factors may be changed (Art. 75), the same as 452 × 3 × 10 × 100. We shall, therefore, find the product if we multiply 452 by 3 and annex three zeros (Art. 79). When the multiplicand and multiplier, or either of them, have zeros at the right hand, the zeros may be disregarded in multiplying, but there must be annexed to the product as many zeros as were disregarded. 26. I have 600 acres in my farm. What is it worth at $250 an acre? 27. How many strawberry plants are there in 400 rows, if there are 280 plants in each row? 28. What is the product of 1870 × 90 ? Of 1870 by 900? Of 1870 by 9000 ? 29. If 268000 is the multiplicand and 80' the multiplier, what is the product? 30. Multiply 596 by 3 and by 40, and add the products. 31. Multiply 984 by 8 and by 60, and add the products. 32. Multiply 647 by 9 and by 20, and add the products. 33. Multiply 379 by 5 and by 80, and add the products. 34. Multiply 4837 by 2, by 30, and by 500, and add the products. 35. Multiply 2802 by 8, by 70, and by 900, and add the products. 81. ILLUSTRATIVE EXAMPLE V. Multiply 625 by 39. 625 39 WRITTEN WORK. = 5625 product by 9. 1875 product by 30. 24375 product by 39. = Explanation. We shall find the product of 625 × 39 if we multiply 625 by 9 and then by 30, and add the results. We first find the product by 9, which is 5625, and write it under the line. The product of 625 × 30 is the same as 625 × 3 × 10. To find this we multiply 625 by 3, obtaining 1875, but instead of annexing a zero (Art. 79), we write the result as 1875 tens. We then add the partial products. NOTE. Compare this process with that of Examples 30 to 33, in the last Article. 36. How many are 34 × 25? 37. Multiply 49 by 98; then multiply 98 by 49. products equal? Why? Are these 38. What is the product of 2842 multiplied by 28? 39. Multiply 3684 by 36 and by 64, and add the products. 40. Multiply 625 by 339; by 705; by 7005. The explanation of this work is left for the pupil. (See Art. 81.) 41. How many are 743 × 657 ? 42. Multiply 237 by 195; 195 by 237. Multiplication of Decimals. 82. ILLUSTRATIVE EXAMPLE VI. Multiply 108.67 by 48. WRITTEN WORK. 108.67 48 86936 43468 = prod. by 8. Explanation.—10867 hundredths multiplied by 8 is 86936 hundredths. 10867 hundredths × 40 is the same as 10867 hundredths × 4 × 10. Now 10867 hundredths × 4 is 43468 hundredths; to express this product multiplied by 10 we write the figures one place to the left. Adding the partial products we have 521616 hundredths (5216.16) for the entire product. Here, as in the preceding examples, we see that the product is of the same order of units as the multiplicand. = prod. by 40. 5216.16 prod. by 48. = 47. Multiply 8.648 by 5; 432.5 by 21. 48. Multiply 7.0909 by 6; 0.0005 by 18. 49. Multiply 0.625 and 0.375 each by 24, and add the results. 83. From the preceding examples may be derived the following Rule for Multiplication. 1. Write the multiplicand and underneath write the multiplier. Draw a line beneath. 2. If the multiplier consists of one term only, multiply each term of the multiplicand by the multiplier, beginning with the term of the lowest order, and carrying as in addition. 3. If the multiplier consists of more than one term, multiply by each term of the multiplier separately, writing the partial products so that units of the same order shall be expressed in the same column. 4. Add the partial products thus obtained, and the result will be the entire product. Proof. Multiply the multiplier by the multiplicand: the two products ought to be equal. For contractions in multiplication, see Appendix, page 300. 84. Examples in Multiplication. a. Multiply 4687 by 8. b. Find the product of 50875 by 7. Ans. 37496. Ans. 356125. c. Multiply 5872 by 10, also by 1000, and add the products. Ans. 5930720. d. Multiply 8756 by 300; by 500; by 7000; and add the results. Ans. 68296800. e. What is the product of 39700 by 9000? g. For multiplicand take 46059, for find the product. Ans. 357300000. Ans. 1608458. multiplier 76, and Ans. 3500484. h. How many are 309 times 46057 ? Ans. 14231613. i. Multiply thirty-seven thousand twenty-eight by 508. Ans. 18810224. j. The multiplier being 987, the multiplicand six thousand four hundred sixteen, required the product. Ans. 6332592. k. What is the product of 908060 by five thousand four hundred ? Ans. 4903524000. 1. One factor being 718151, the other seven hundred, what is the product? Ans. 502705700. m. At 147 dollars per acre, how much will 385 acres of land cost? n. There are 24 hours in a day. 476 days? Ans. $56595. How many hours in Ans. 11424. Examples with Decimals. o. Multiply 40.27 by 87. p. Multiply thirty-one thousandths by 25. Ans. 3503.49. Ans. 0.775. NOTE. The examples on this page embrace the chief varieties in form of examples in Multiplication. Examples for additional practice will be found on pages 59-63. 85. Applications. 50. At $45 a month for labor, what will a man earn in a year? In 5 years? 51. If a man saves $17 a month, what will he save in 25 years? 52. If a sewing-machine can set 690 stitches in a minute, how many stitches can it set in 60 minutes or an hour? In a day of 12 hours? In 6 working days or a week? In 52 weeks or a year? 53. The first House of Representatives of the United States consisted of 65 members; if each member represented 30000 inhabitants, how many inhabitants were represented? 54. In a certain mill, material for 65000 dresses is made in a week. Allowing 18 yards for a dress, how many yards are made in a week? In a year? 55. The cotton crop in Texas in one year was 450000 bales. Allowing 400 pounds to a bale, how many pounds were raised? 56. In a day there are 24 hours, in an hour 60 minutes, in a minute 60 seconds. How many seconds in a day? 57. Light, according to Foucault, travels at the rate of 185172 miles in a second. If it passes from the sun to the earth in 8 minutes 13 seconds (or 493 seconds), what is the distance from the sun to the earth? 86. Examples with Decimals. 58. It took Mary 3.25 hours to learn a piece of music, and Olive 5 times as long. How many hours was Olive in learning it? 59. Mr. Green has 5.175 acres of land and buys 7 times as much of his neighbor. How many acres does he buy of his neighbor? 60. What will 38 barrels of flour cost at $11.75 a barrel? 61. Mr. Gage sold 175 tons of refined bar-iron at $45.50 a ton. What did he receive for it? 62. Multiply 5.4328 by 62. |