plier under it, we may say, 5 times 7 are 35, writing down the 5, and reserving the 3 (tens) as in addition. Again, 5 Multiplicand, 27 trees in each row. times 2 (tens) are 10, (tens,) and 3, (tens,) which we reserved, make 13, (tens,) as before. 10. 12. There are on a board 3 rows of spots, and 4 spots in each row; how many spots on the board? A slight inspection of the figure will show, that the number of spots may be found either by taking 4 three times, (3 times 4 are 12,) or by taking 3 four times, (4 times 3 are 12;) for we may say there are 3 rows of 4 spots each, or 4 rows of 3 spots each; therefore, we may use either of the given numbers for a multiplier, as best suits our convenience. We generally write the numbers as in subtraction, the larger uppermost, with units under units, tens under tens, &c. Thus, Multiplicand, 4 spots. Product, 12 Ans. Note. 4 and 3 are the factors, which produce the product 12. Hence, Multiplication is a short way of performing many additions; in other words,-It is the method of repeating any number any given number of times. SIGN. Two short lines, crossing each other in the form of the letter X, are the sign of multiplication. Thus, 3 X 4 12, signifies that 3 times 4 are equal to 12, or 4 times 3 are 12. Note. Before any progress can be made in this rule, the following table must be committed perfectly to memory. 13. What will 84 barrels of flour cost at 7 dollars a barrel? Ans. 588 dollars. 14. A merchant bought 273 hats at 8 dollars each; what did they cost? Ans. 2184 dollars. 15. How many inches are there in 253 feet, every foot being 12 inches? OPERATION. 253 12 The product of 12, with each of the significant figures or digits, having been committed to memory from the multiplication table, it is just as easy to multiply by 12 as by a Ans. 3030 single figure. Thus, 12 times 2 are 36, &c. 16. What will 476 barrels of fish cost at 11 dollars a barrel? Ans. 5236 dollars. 17. A piece of valuable land, containing 33 acres, was sold for 246 dollars an acre; what did the whole 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 multiplier exceeds 12, we multiply by each figure in the multiplier.separately. Thus : OPERATION. 246 dollars, the price of 1 acre. 738 dollars, the price of 3 acres. Ans. 8118 dollars, the price of 33 acres. The multiplier consists of 3 tens and 3 units. First, multiplying by the a units gives u 738 dollars, the price of 3 acres. We then multiply by the 3 tens, writing the first figure of the product (8) in ten's place, that is, directly under the figure by which we multiply. It now appears, that the product by the 3 tens consists of the same 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, by which their value is increased tenfold, which is as it should be, because 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 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 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 1 day, at that rate? how many miles in 36 days? how many miles in 1 year, or 355 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 315,) 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 × 424.) 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 × 420 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 × 3 = 60 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 component 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 received 8X7X2=112. Ans. 38304 dollars. |