9. 4627381x1167000. 10. 7854392 × 4800000. 11. 5869x305. 12. 5086×508. 13. 7402x1008. 14. 80024x8104. § 7. CASE 2. When the multiplier is the product of two or more numbers, or a composite number, greater than 12. RULE. Multiply by each of those parts separately, instead of the whole number at once. The reason of this method is obvious; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number: thus 9 times a given number multiplied by 8, gives 72 times the same number, as clearly as 9 times 8 make 72. EXAMPLES. 1. Multiply 213789465 by 36, or by 6 times 6. § 8. CASE 3. To multiply by any number of nines in one line, as 9, 99, 999, &c. RULE. Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from this number subtract the multiplicand, the difference will be the product required. § 9. CASE 4. To find the product of one or more figures from the product of a figure already found. RULE. Multiply the product found by that number which, when multiplied by the given number, produces the other part of the multiplier. In the 1st example the product of 4 is multiplied by 2, in the 2d, the product of 9 is multiplied by 4; and in the 3d, the product of 7, (or rather 700,) is multiplied by 9. 4. Multiply 36284 by 54279. | 7. Multiply 74820 by 27030. 284196324.8. 66 42061 " 66 6183.9. 66 5614236484. 17864 " 3612. Q. How do you multiply integers when the multiplier consists of several figures? Q. How do you multiply, when ciphers are annexed to any or both of the factors? Q. How do you multiply, when the multiplier is a com posite number? Q. How do you prove multiplication? MISCELLANEOUS EXERCISES. 1. Multiply 41152263 by 2. | 8. Multiply 771604932 by 16. 15. Moscow contains 305631 inhabitants. If there were sixteen times as many, what number would there be? 16. The university of Moscow had in 1835, 120 professors, and 430 pupils. If there were 430 pupils for each professor, how many would there be? 17. If one acre of well tilled land produces 218 bushels of potatoes, how many bushels will 573 acres produce? 18. For every ox sold in a certain market there are sold 98 sheep. If 74184 oxen were disposed of, how many sheep would be sold? 19. If 383042 pounds of lead are sold in one year, how many pounds will be sold in 9 years? 20. If a man can walk 9875 miles in a year, how far could he walk in 16 years? 21. If one house cost 1263 dollars, how much will a row of 13 houses cost? 22. If one calf be worth 11 dollars, how many dollars would 14303 calves cost? 23. The population of Brussels in 1837, was about 134265. If Spain contains 33 times as many inhabitants as thecity of Brussels, what is the number of its population? 24. From Charleston to Madeira is 3600 miles. Over how many miles does a ship sail in making 37 voyages between the two places? 25. From Norfolk to Portugal is 3700 miles. From New York to Portugal is 3400 miles. How many more miles will a ship sail in making 19 voyages between Norfolk and Portugal, than another vessel which makes the same number of voyages between that country and New York? 26. How much must a manufacturer receive for 695876 yards of cloth, at 7 dollars per yard? 27. A grocer sold 4569 cwt. of sugar, at the rate of 9 dol lars per cwt., how many dollars did he receive? 28. What is the value of 13487 barrels of flour at 6 dollars per barrel? 29. In the District of Columbia there are 100 square miles, and on each mile 437 inhabitants; what is the population? 30. What will be the cost of building a ship of 576 tons, at 56 dollars per ton? 31. There are 365 days in a year; how many in 4395 years? 32. Bought a farm of 267 acres, at the rate of 75 dollars per acre; what did it cost? 33. A planter sold to a merchant, on his note at 6 months, 3297 bales of cotton, at 75 dollars per bale; what was the amount of the note? 34. What is the content of a rectangular piece of ground whose length is 465, and breadth 95 perches? 35. There are 1760 yards in a mile; how many yards in 458 miles? 36. To square a number, is to multiply the number by itself; what is the square of 6045? 37. How many cubic inches in 47095 wine gallons, each gallon containing 231 inches? 38. At New Gibraltar, of a still night, the sentinel's cry of "All's well," can be heard in 55 seconds after it has been uttered at Old Gibraltar; required the distance between the two places, sound moving at the rate of 1142 feet in a second? 39. What will 143 head of cattle come to, at 19 dollars per head? 40. There are 640 acres in one square mile, how many acres in 69075 square miles? 41. What is the cost of constructing 609 miles of rail road, at 15085 dollars a mile? 42. How many square feet of boards in a floor 72 feet long and 45 feet wide? 43. If 36 men can perform a piece of work in 159 days, how long will it take one man? 44. How many times will a wheel turn in 1475 miles, supposing it to turn 320 times in one mile? 45. At 225 dollars per ton, what will 498 tons of hemp cost? 46. The earth casts a shadow whose mean length is equal to 108 times the diameter; required the length of the shadow, the diameter being 7935 miles ? D 47. To cube a number is to multiply the square of the number by the number itself: what is the cube of 625 ? 48. A steamer moves at the rate of 16 miles an hour; what distance will she pass over in 24 hours? 49. How many feet of earth can be removed in 35 days, by a cart carrying 15 loads a day and 36 feet at a load? 50. A merchant purchased 34 bales of cloth, each bale contained 23 pieces, and each piece 27 yards. How many pieces and yards, and what is the value of the whole at 5 dollars per yard. 51. How many dollars are equal to 23 bags, each containing 643 doubloons, the doubloon being valued at 16 dollars. 52. The pressure of the atmosphere upon a square inch is 15 pounds, what is the pressure upon a square foot which contains 144 inches? 53. How many ounces of provisions must a ship's company of 525 men have to last 93 days, allowing to each man 16 ounces per day? 54. To dig a canal 137280 yards long, 48 yards wide, and 3 yards deep, how many solid yards of earth must be removed? 55. It has been estimated that the river Ganges, during the flood season, discharges daily into the sea a mass of mud equal to 74 times the weight of the great Egyptian pyramid. Required the quantity discharged in 26 days, the weight of the pyramid being 2164956550400 pounds. DIVISION. § 10. Division is that operation by which we find how often one number is contained in another, and is a short method of performing subtraction. Q. What does division mean? A. It means the separating into equal parts: thus, two numbers are given, one of which is to be divided into parts; the other shows how many parts there are to be; and the object of the operation is to find the value of one of those parts. Q. Do these numbers get particular names? A. The number to be divided is called the dividend; the number you divide by is called the divisor, and shows how many parts there are; and the number of times the divisor is |