PROOF OF MULTIPLICATION. Write the multiplicand in the place of the multiplier, and find the product as before: if the two products are the same, the work is supposed right. Q. When the multiplier exceeds 12, how do you set it down? How do you multiply by it? How do you add up? How many numbers are there in every multiplication? Name them? How do you prove multiplication? EXAMPLES. 1. Multiply 365 by 84: also, 37864 by 209. (1.) (3.) 34293 (4.) 47042 Multiplicand 365 (2.) 37864 Multiplier 84 209 11. Multiply 1345894 by 49. 12. Multiply 576784 by 64. 19. Multiply 86972 by 1208. Ans. 10264849920. Ans. 6242102428164 Ans. Ans. 601380780. Ans. 405768300. Ans. 12276971600. CASE III. 25. When the multiplier is 1 and any number of ciphers after it, as 10, 100, 1000, &c. Placing a cipher on the right of a number changes the units place into tens, the tens into hundreds, the hundreds into thousands, &c., and therefore increases the number ten times. So Thus, 5 is increased ten times by making it 50. the addition of two ciphers increases a number one hundred times; the addition of three ciphers, a thousand times, &c. Thus, 6 is increased a hundred times by making it 600, and 5 is increased a thousand times, by making it 5000. Hence, we have the following RULE. Place on the right of the multiplicand as many ciphers as there are in the multiplier, and the number so formed will be the required product. Q. 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? And for any number of ciphers, how much will each increase it? How do you multiply by 10, 100, 1000, &c.? § 26. When there are ciphers on the right hand of one or both of the factors. RULE. Neglect the ciphers and multiply the significant figures: then place as many ciphers to the right hand of the product, as there are in both of the factors. Q. When there are cipners on the right hand of both the factors, how do you multiply? § 27. When the multiplier is a composite number. A composite number is one that may be produced by the multiplication of two or more numbers, which are called the components or factors. Thus, 2x3=6. Here 6 is the composite number, and 2 and 3 are the factors, or components. The number 16-8×2: here 16 is a composite number, and 8 and 2 are the factors; and since 4x4-16, we may also regard 4 and 4 as factors or components of 16. Q. What is a composite number? Is 6 a composite number? What are its components or factors? What are the factors of the composite number 16? What are the factors of the composite number 12? EXAMPLES. 1. Let it be required to multiply 8 by the composite number 6, in which the factors are 2 and 3. If we write 6 horizontal lines with 8 units in each, it. is evident that the product of 8×6=48, the number of units in all the lines. But let us first connect the lines in sets of 2 each, as on the right; there will then be in each set 8x2=16; or 16 units in each set. But there are 3 sets; hence, 16x3=48, the number of units in all the sets. If we divide the lines into sets of 3 each, as on the left, the number of units in each set will be equal to 8×3=24, and there being 2 sets, 24×2=48, the whole number of units. As the same may be shown for all numbers we have the following RULE. When the multiplier is a composite number, multiply by each of the factors in succession, and the last product will be the entire product sought. EXAMPLES. 1. Multiply 327 by 12. The factors of 12, are 2 and 6, or they are 3 and 4, or they are 3, 2 and 2: for, 2×6=12, 3×4=12, and 2. Multiply 5709 by 48; the factors being 8 and 6, or 1. There are ten bags of coffee, each containing 48 pounds: how much coffee is there in all the bags ? Ans. lbs. 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? Ans. 4415 yards. Ans. hours. 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. A merchant bought 49 hogsheads of molasses, each containing 63 gallons: how many gallons of molasses were there in the parcel ? Ans. gallons. 7. Suppose a man were to travel 32 miles a day: how far would he travel in 365 days? Ans. 11680 miles. 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 town? Ans. inhabitants. 9. 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 usually called Bills of Parcels. James Johnson BILLS OF PARCELS. New-York, Oct. 1, 1838. 4 Chests of tea, of 45 pounds each, at I 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, Amount 1211 dollars. W. Smith. |