is evidently ten times as much as the price of 3 acres, that is, 7380 dollars; and it is plain, that these two products, added together, give the price of 33 acres. These examples will be sufficient to establish the following RULE. I. Write down the multiplicand, under which write the multiplier, placing units under units, tens under tens, &c., and draw a line underneath. II. When the multiplier does not exceed 12, begin at the right hand of the multiplicand, and multiply each figure contained in it by the multiplier, setting down, and carrying, as in addition. III. When the multiplier exceeds 12, multiply by each figure of the multiplier separately, first by the units, then by the tens, &c., remembering always to place the first figure of each product directly under the figure by which you multiply. Having gone through in this manner with each figure in the multiplier, add their several products together, and the sum of them will be the product required. EXAMPLES FOR PRACTICE. 18. There are 320 rods in a mile; how many rods are here 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 1 day, at that rate? how many miles in 36 days? how many miles in 1 year, or 365 days? Ans. 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 37864 by 235. CONTRACTIONS IN MULTIPLICATION. I. When the multiplier is a composite number. 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 multiplication 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 factors of that number. So, also, 24 is a composite number; its component parts or factors may be 2 and 12 (2 × 12=24;) or they may be 4 and 6 (4 × 6 24;) or they may be 2, 3, and 4 (2 × 3 × 4—24.) 4 5 20 3 60 = 1. What will 15 yards of cloth cost, at 4 dollars a yard? 15 yards are equal to 5 × 3 yards. The cost of 5 yards would be 5 X 420 dollars; and because 15 yards contain 3 times 5 yards, so the cost of 15 yard will evidently be 3 times the cost of 5 yards, that is 20 dollars X 360 dollars. Ans. 60 dollars Wherefore, If the multiplier be a composite number, we may if we please, multiply the multiplicand first by one of the com ponent parts, that product by the other, and so on, if the component parts be more than two; and, having in this way multiplied by each of the component parts, the last product will be the product required. 2. What will 136 tons of potashes come to, at 96 dollars per ton ? 8 X 1296. 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 component 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 to receive 112 dollars each, how much will they all receive? 8 X 7 X 2 = 112. Ans. 38304 dollars. II. When the multiplier is 10, 100, 1000, &c. 78192. ¶ 12. It will be recollected, (T3.) that any figure, on being removed one place towards the left hand, has its value increased tenfold; hence, to multiply any number by 10, 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 tens, 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 multi plicand, so increased, will be the product required. Thus, Multiply 46 by 10, the product is 460. EXAMPLES FOR PRACTICE. 1. What will 76 barrels of flour cost, at 10 dollars a barrel? 2. If 100 men receive 126 dollars each, how many dollars will they all receive? 3. What will 1000 pieces of broadcloth cost, estimating each piece at 312 dollars? 4. Multiply 5682 by 10000. 5. T 82134 ... 100000. 13. 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 to the product as there are ciphers on the right hand of the multiplicana and multiplier, counted together. EXAMPLES FOR PRACTICE. 1. If 1300 men receive 460 dollars apiece, how many dollars will they all receive? OPERATION. 460 1300 138 46 Ans. 598000 dollars. The ciphers in the multiplicand and multiplier, counted together, are three. Disregarding these, we write the significant figures of the multiplier under the significant figures of the multiplicand, and multiply; after which we annex three ciphers to the right hand of the 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 1826, at 400, running, on an average, 700 spindles each; what was the whole number of spindles ? 3. Multiply 357 by 6300. 1512 000 756 77112 In the operation it will be seen, that mult plying by ciphers produces nothing. There fore, 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 directly under the figure by which we multiply. EXAMPLES FOR PRACTICE. 8. Multiply 154326 by 3007. 1. What is multiplication? 2. What is the number to be multiplied 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 number be placed for multiplication? 9. How do you proceed when the multiplier 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 suffices? 15. Why? 16. When there are ciphers on the right hand of the multiplicand, multiplier, either or both, how may we proceed? 17. When there are ciphers 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, that, when it was shared among them, each man received 46 dollars; what was the sum of money taken ? |