OPERATION, OPERATION. 13. What will 84 barrels of four cost at 7 dollars a barrel ? Ans. 5.98 dollars. 14. A merchant bought 273 hats at 8 dollars each; what did they cost ? Ans. 21:34 doilars. 15. How many inches are there in 253 feet, every fool being 12 inches ? The product of 12, with each of the significant 253 figures or digits, having been comm. led to -12 memory from the multiplication table, it is just as easy to inultiply by 12 as by a single tigure. Ans. 3036 Thus, 12 times 3 are 36, &c. 16. What will 476 barrels of fish cost at 11 dollars a harrel ? Ans. 5236 dollars, 17. A piece of valuable land, containing 33 acres, was soid for 246 dollars an acre ; what did the whcle come to? As 12 is the largest number, the product of which, with the nine digits, is found in the multiplication table, therefore, when the mula tiplier exceeds 12, we multiply by each figure in the multiplier separately. Thus: The multiplier consists 246 dollars, the price of 1 acre. of 3 tens and 3 units. 33 number of acres. First, multiplying hy the 3 units gives us 738 733 dollars, the price of 3 acres. dollars, the price of 3 738 dollars, the price of 30 acres. We ihen mul. tiply by the 3 tens, Ans. 8118 dollars, the price of 33 acres. writing the first figure of the product (3) in ten's place, that is, directly under the figure by which we multiply. Il now appears, that the product hy the 3 lens consists of the saine figures as the product by the three units; but there is this difference--the figures in the product by the 3 tens are all removed one place further toward the left hand, loy which their value is increased ten fold, which is as it should he, hecause the price of 30 acres is evidently ten times as much as the price of 3 acres, that is, 7380 dollars; and it is plain, that these iwo products, added together, give the price of 33 acres. These examples will be sufficient to establish the following RULE. 1. Write down the multiplicand, under which write the multiplier, placing units under units, tens under tens, &c., and draw a line underneath. M. When the multiplier does not exceed 12, begin at the right hand of the multiplicana, and multiply each figure contained in it by the multiplier, setting down, and carrying, as in addition. acres. III. When the multiplier e.treeds 12, multiply hy each figure of the multiplier separately, first by the units, then hy'the tens, &c., remembering always to place the first figure of each product directly under the figure hy which you multiply. Having gone through in this manner with each figure in the multiplier, and their several products together, and the sum of them will be the product required. EXAMPLES FOR PRACTICE. 18. There are 320 rods in a inile; how many rods are there in 57 miles ? 19. It is 436 miles from Boston to the city of Washington; how many rods is it? 20. What will 784 chests of tea cost, at 69 dollars a chest ? 21. If 1851 men receive 758 dollars apiece, how many dollars will they all receive ? Ans. 1403058 dollars. 22. There are 24 hours in a day; if a ship sail 7 miles in an hour, how many miles will she sail in one day, at that rate ? how many miles in 36 days ? how many miles in 1 year, or 365 days? Åns. 61320 miles in 1 year. 23. A merchant bought 13 pieces of cloth, each piece containing 28 yards, at 6 dollars a yard; how many yards were there, and what was the whole cost ? 'Ans. The whole cost was 2184 dollars. 24. Multiply 37861 hy 235. Product, 8898040. 25. 29831 952. 28399112. 8704. 817793021. . 26. 93956 . CONTRACTIONS IN MULTIPLICATION. I. When the multiplier is a composile number.. T 11. Any number, which may be produced by the multiplication of two or more numbers. is called a composite 'number. Thus, 15, which arises from the inultiplication of 5 and 3, (5 X 3=15,) is a composite number, and the numbers 5 and 3, which, multiplied together, produce it, are called component parts, or fuctors of that number. So, also, 24 is a composite number; its component parts or factors may be 2 and 12, (2 X 12 = 24,) or they may be 4 and 6, (4 X6 = 24,) or they may be 2, 3, and 4, (2 X 3 X 4 = 24.) 1. What will 15 yards of cloth cost at 4 dollars a yard ? 4 5 20 3 15 yards are equal to 5 X 3 yards. The cost of 5 yards would he 5 X 4 = 20 dollars; and because 15 yards contain 3 times 5 yards, so the cost of 15 yards will evidently be 3 times the cost of 5 yards, that is, 20 dollars X 3 = 60 doilars. Ans. 60 dollars. 60 Wherefore, If the nulliplier be a composite number, we may, if we please. multiply the multiplicand first by one of the component parts, that product by the other, and so on, is the component parts be more ihan two; and, having in this way multiplied by each of the compobeat parts, the last product will lie the product required. 2. What will 136 tons of polashes come to, at 90 dollars per ton ? 8 X 12 = 96. It follows, therefore, that 8 and 12 are component parts or factors of 96. Hence, 136 dollars, the price of 1 ton. 8 one of the component parts, or factors. 1088 dollars, the price of 8 tons. 12 the other coinponent part, or factor. Ans. 13056 dollars, the price of 96 tons. 3. Supposing 342 men to be employed in a certain piece of work, for which they are lo receive 112 dollars each, how much will they all receive ? 8 X 7 X 2 = 112. Ans. 38304 dollars, 4. Multiply 367 by 48. Product, 17616. 5. 853 56. 47763. 6. 1086 72 78192. II. When the multiplier is 10, 100, 1000, &c. T 12. It will be recollected, (1 3,) that any figure, on being removed one place towarıis the left hand, has its value increased ten fold; hence, to multiply any number loy 10, it is only necessary to write a cipher on the right hand of it. Thus, 10 times 25 are 250 ; for the 5, which was units before, is now made lens, and the 2, which was tens before, is now made hundreds. So, also, if any figure be removed two places towards the left hand, its value is increased 100 times, &c. Hence, When the multiplier is 10, 100, 1000, or 1 with any number of ciphers annexed, annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the multiplicand, so increased, will be the prorluct required. Thus, Multiply 46 by 10, the product is 460. 8300. . 95000. EXAMPLES FOR PRACTICE. 1. What will 76 harrels of four cost, at 10 dollars a barrel ? 2. If 100 men receive 126 dollars each, how many dollars will hey all receive ? 3. What will 1000 pieces of broadcloth cost, estimating each piece at 312 dollars ? 4. Multiply 5682 hy 10000. 5. 82134. 100000. 113. On the principle suggested in the last T, it follows, When there are ciphers on the right hand of the multiplicand, multiplier, either or both, we may, at first, neglect these ciphers, multiplying by the significant figures only; after which we must annex as many ciphers io the product as ihere are ciphers on the right hand of the multiplicand and multiplier, counted iogether. EXAMPLES FOR PRACTICE. 1. If 1300 men receive 400 dollars apiece, how many dollars will they all receive ? annex The ciphers in the multiplicand and OPERATION. multiplier, counted together, are three. 460 Disregarding these, we write the signifi1300 cant figures of the multiplier under the significant figures of the multiplicand, 138 and multiply; after which we 46 three ciphers to the right hand of the Ans. 598000 dollars. product, which gives the true answer. 2. The number of distinct buildings in New England, appropriated to the spinning, weaving, and printing of cotton goods, was estimated, in 18-26, at 400, running, on an average, 700 spindles each; what was the whole number of spindles ? 3. Multiply 357 by 6300. 4. 8600 17. 5. 9340 460. 6. 5200 410. 7. 204. . . . 378 . . OPERATION. 378 204 In the operation it will be seen, that multiplying by ciphers produces nothing. Therefore, 1512 000 756 77112 III. When there are ciphers between the significant figures of the multiplier, we may omit the ciphers, multiplying by the significant figures only, placing the first figure of each product direcily under the figure by which we multiply. EXAMPLES FOR PRACTICE. 8. Multiply 154326 by 3007. OPERATION. 154326 3007 1080282 462978 Product, 464058282 9. Multiply 543 by 206. 10. 2103, 11. 36243 32004. 1620 SUPPLEMENT TO MULTIPLICATION. QUESTIONS. 1. What is multiplication? 2. What is the number to be multi called ? 3. - to multiply by called? 4. What is the result or answer called? 5. Taken collectively, what are the multiplicand and multiplier called ? 6. What is the sign of multiplication ? 7. What does it show? 8. In what order must the given numbers le placed for multiplication ? 9. How do you proceed when the mul. tiplier is less than 12 ? 10. When it exceeds 12, what is the method of procedure ? 11. What is a composite number? 12. What is to be understood by the component parts, or factors, of any number? 13. How may you proceed when the multiplier is a composite number? 14. To multiply by 10, 100, 1000, &c., what suihices? 15. Why? 16. When there are ciphers on the right hund of the multiplicand, multiplier, either or both, how may we proceed? 17. When there are cipliers between the significant figures of the multiplier, how are they to be treated ? EXERCISES. 1. An army of 10700 men, having plundered a city, took so much money thai, when it was shared amoig ihem, each man received 46 dollars; what was the sum of money taken ? 2. Supposing the number of houses in a certain town to be 145, cach house, on an average, containing two fainilies, and each family 6 members, what would be the number of inhabitants in that lown? Ans. 1740. 3. If 46 men can do a piece of work in 60 days, how many men will it take to do it in one day? 4. Two men depart from the same place, and travel in opposite directious, one at the rate of 27 iniles a day, the other 31 miles a day: \ how far apart will they le at the end of 6 days ? Ans. 348 miles. 5. What number is ihat the factors of which are 4, 7, 6, and 20 ? Ans. 3360. 6. If 13 men can do a piece of work in 90 days, how long will it take one man to do the saine? 7. What sum of money must be divided between 27 men, so that each mau may receive 115 dollars ? 8. There is a certain number, the factors of which are 89 and 265; what is that nuinler ? 9. What is that number, of which 9, 12. and 14 are factors ? 10. If a carriage whecl turn round' 346 times in running 1 inile, how many times will it turn round in the distance from New York to Philadelphia, it being 95 miles ? Ans. 32370. 11. In one minute are 60 seconds; how many seconds in 4 minutes ? in 5 minutes ? in 20 minutes ? - in 40 minutes ? 12. In one hour are 60 minutes; how many seconds in an hour ? in two hours ? how many seconds froin nine o'clock in the morning till noon ? 13. In one dollar are 6 shillings; how many shillings in 3 dollars ? in 300 dollars ? in 467 dollars ? 14. Two men, A and B, start from the same place at the same time, and travel the same way; A travels 52 miles a day, and B 44 miles a day; how far apart will they le at the end of lv days? 15. If the interest of 100 cents, for one year, be 6 cents, how many cents will be the interest for 2 years ? for 4 years? for 10 years? for 35 years ? for 84 years ? |