Again, if we divide the lines into sets of 3 each, as at the left, the number of units in each set will be equal to 8x3=24, and since there are two sets, the whole number of units will be expressed by 24 × 2=48. Since the product of either two of the three factors 8, 3 and 2, will be the same whichever be taken for the multiplier (48), and since the same principle will apply to that product and the other factor, as well as to any additional factor, if introduced, it follows that, The product of any number of factors will be the same in whatever order they are multiplied: hence, the following RULE.-I. Separate the composite number into its factors. II. Multiply the multiplicand and the partial products by the factors, in succession, and the last product will be the entire product sought. 1. Multiply 327 by 12. EXAMPLES. The factors of 12 are 2 and 6; they are also 3 and 4 ; or they are 3, 2 and 2. For, 2×6=12, 3 × 4=12, and 3 × 2 × 2=12. 2. Multiply 5709 by 48. 5. Multiply 937387 by 54. 6. Multiply 91738 by 81. 7. Multiply 3842 by 144. CASE II. 55. When the multiplier is 1, with any number of ciphers annexed, as 10, 100, 1000, &c. Placing a cipher on the right of a number, is called annexing it. Annexing one cipher increases the unit of each place ten times: that is, it changes units into tens, tens into hundreds, hundreds into thousands, &c.; and therefore increases the number ten times. Thus, the number 5 is increased ten times by annexing one cipher, which makes it 50. The annexing of two ciphers 55. If you place one cipher on the right of a number, what effect has it on its value? If you place two, what effect has it? If you place three? How much will each increase it? How do you multiply by 10, 100, 1000, &c ? increases a number one hundred times; the annexing of three ciphers, a thousand times, &c.: hence the following RULE.-Annex to the multiplicand as many ciphers as there are in the multiplier, and the number so formed will be the required product. EXAMPLES. 1. Multiply 254 by 10. 5. Multiply 3750 by 100. CASE III. 56. When there are ciphers on the right hand of one or both of the factors. In this case each number may be regarded as a composite number, of which the significant figures are one factor, and 1, with the requisite number of ciphers annexed, the other. 1. Let it be required to multiply 3200 by 800. OPERATION. 3200=32 × 100; and 800-8 × 100; Hence, we have the following RULE.-Omit the ciphers and multiply the significant figures: then place as many ciphers at the right hand of the product as there are in both factors. APPLICATIONS IN MULTIPLICATION. 57. The analysis of a practical question, in Multiplication, requires that the multiplier be an abstract number; and then the unit of the product will be the same as the unit of the multiplicand. Thus, what will 5 yards of cloth cost at 7 dollars a yard? ANALYSIS.-Five yards of cloth will cost 5 times as much as 1 yard. Since 1 yard of cloth costs 7 dollars, 5 yards will cost 5 times 7 dollars, which are 35 dollars. The cost of any number of things is equal to the price of a single thing multiplied by the number. But we have seen that the product of two numbers will be the same, (that is, will contain the same number of units) whichever be taken for the multiplicand (Art. 48). Hence, in practice, we may multiply the two factors together, taking either for the multiplier, and than assign the proper unit to the product, We generally take the least number for the multiplier. QUESTIONS FOR PRACTICE. 1. There are ten bags of coffee, each containing 48 pounds: how much coffee is there in all the bags ? 2. There are 20 pieces of cloth, each containing 37 yards, and 49 other pieces, each containing 75 yards: how many yards of cloth are there in all the pieces? 3. There are 24 hours in a day, and 7 days in a week: how many hours in a week? 4. A merchant buys a piece of cloth containing 97 yards, at 3 dollars a yard: what does the piece cost him? 5. A farmer bought a farm containing 10 fields; three of the fields contained 9 acres each; three other of the fields 12 acres each; and the remaining 4 fields each 15 acres: how many acres were there in the farm, and how much did the whole cost at 18 dollars an acre? 6. Suppose a man were to travel 32 miles a day: how far would he travel in 365 days? 56. When there are ciphers on the right hand of one or both the factors, how do you multiply? 57. What does the analysis of a practical question require? How do you find the cost of a single thing? How may it be done in practice? 7. A merchant bought 49 hogsheads of molasses, each containing 63 gallons: how many gallons of molasses were there in the parcel ? 8. In a certain city there are 3751 houses. If each house on an average contains 5 persons, how many inhabitants are there in the city? 9. If a regiment of soldiers contains 1128 men, how many men are there in an army of 106 regiments? 10. If 786 yards of cloth can be made in one day, how many yards can be made in 1252 days? 11. If 30009 cents are paid for one man's labor on a railroad for 1 year, how many cents would be paid to 814 men, each man receiving the same wages? 12. There are 320 rods in a mile; how many rods are there in the distance from St. Louis to New Orleans, which. is 1092 miles? 13. Suppose a book to contain 470 pages, 45 lines on each page, and 50 letters in each line: how many letters in the book? 14. Supposing a crew of 250 men to have provisions for 30 days, allowing each man 20 ounces a day: how many ounces have they? 15. There are 350 rows of trees in a large orchard, 125 trees in each row, and 3000 apples on each tree: how many apples in the orchard? 16. What is the cost of 7585 barrels of flour at 7 dollars a barrel? 17. If a railroad car goes 27 miles an hour, how far will it run in 3 days, running 20 hours each day? How far would it run if its rate were 37 miles an hour? 18. If 1327 barrels of flour will feed the inhabitants of a city for 1 day, how many barrels will supply them for 2 years? 19. A regiment of men contains 10 companies, each company 8 platoons, and each platoon 34 men: how many men in the regiment? 20. Two persons start from the same place and travel in the same direction: one travels at the rate of 6 miles an hour, the other at the rate of 9 miles an hour. If they travel 8 hours a day, how far will they be apart at the end of 17 days? How far if they travel in opposite directions? 21. The Erie railroad is about 425 miles long, and cost 65 thousand dollars a mile: what was the entire cost of construction? 22. A drover bought 106 oxen at 35 dollars a head; it cost him 6 dollars a head to get them to market, where he sold them at 47 dollars; did he make or lose, and how much? 23. The great Illinois Central Railroad reaches from Chicago to the mouth of the Ohio river, 815 miles: it cost 23500 dollars a mile: what was its entire cost? 24. Mr. Denning's orchard is square and contains 36 trees in a row each tree yields 4 barrels of appies which he sells for 2 dollars a barrel: how much does he get for his crop? BILLS OF PARCELS. 58. When a person sells goods he generally gives with them a bill, showing the amount charged for them, and acknowledging the receipt of the money paid; such bills are called Bills of Parcels. 25 James Johnson, New York, Oct. 1, 1854. Bought of W. Smith. 4 Chests of tea, of 45 pounds each, at 1 doll, a pound. 3 Firkins of butter at 17 dolls. per firkin 4 Boxes of raisins at 3 dolls. per box 36 Bags of coffee at 16 dolls. each 14 Hogsheads of molasses at 28 dolls. each Received the amount in full. 26 James Hughes, Amount, dollars. W. Smith Hartford, Nov. 1, 1854. 27 Bags of coffee at 14 dollars per bag 59 Hogsheads of molasses at 29 dolls. per hogshead, Amount, Received the amount in full, for W. Jones, dollars. per James Cross. 58. What are bills of parcels? |