12. In the city of Pekin in China, is a bell weighing, it is said, 120,000 lbs. ; and at Nankin, in the same country, is another weighing 50,000lbs. ; the first exceeds the great bell at Erfort, in Upper Saxony, by 94,600lbs. ; how much then is the German bell inferior in weight to the second? Ans. 24,600lbs. MULTIPLICATION OF SIMPLE NUMBERS, COMMONLY CALLED SIMPLE MULTIPLICATION. 31. SIMPLE MULTIPLICATION is repeating any given simple number as often as there are UNITS in another; and is a short method of performing ADDITION. BY ADDITION, Illus. 3 boys had 37-cents each; how many had they all? 32. This sign × signifies MULTIPLICATION, and is read mul. tiplied by, or into. As 3x5=15, that is to say, 3 multiplied by 5, or 3 into 5 is equal to 15. 33. There are THREE terms used in multiplication, namely, the multiplicand, multiplier, and product. I. The MULTIPLICAND is the number given to be multiplied. Illus. Oliver bought 3 books at 49 cents apiece; how much did they come to ? PACTORS. cents. Multiplicand 4940+9 3 no. of books. 27 9 × 3 12.40 x 3. Product 147 = 49 X 3 But this process may be abridged: Thus-49 3 Prod. 147 Here we multiply the 9 (units) by the 3, saying 3 times 9 is 27; set down 7, and carry 2. Then the 4 (tens,) saying 3 times 4 is 12, and 2 we had to carry is 14; set down 14. NOTE. As the learner has committed the Table, it will be as easy to multiply by 12, as by any single figure, and the same process is to be observed as in this example. Hence, 34. To multiply simple numbers, (When the multiplier does not exceed 12,) RULE. Multiply EACH FIGURE in the multiplicand by the multiplier, observing to set down and carry as in ADDITION. EXAMPLES. 1. James bought 5 yards of linen at 57 cents a yard; how much did it come to ? Ans. 285 cents. Ans. 126 dollars. 2. What will 7 cows come to at 18 dollars a head? Ans. 608 cents. 4. Robert sold 9 muskrat skins at 37 cents apiece, what did he get for them? Ans. 333 cents. 5. What will 6 pounds of butter come to at 18 pence a pound? Ans. 108 pence. 6. What will 27 pounds of sugar come to at 11 cents a pound? Ans. 297 cents. 7. At 10 cents a yard, what will 29 yards of cotton cloth come to? Ans. 290 cents. 8. A barrel of beer contains 36 gallons, and each gallon 4 quarts; how much, then, will it come to at 12 cents a quart? Ans. 1728 cents. Oliver bought 33 books at 49 cents apiece; how much did they come to? Multiplicand Multiplier Prod. 49 33 FACTORS. 14749 X 3 147.49 × 30 161749 X 33 First, we multiply by the 3 (units) as in Illus. Art. 33; then by the 3 tens, saying 3 times 9 is 27 (tens): we set down 7 (the unit figure of this product) in ten's place, which is, directly under the MULTIPLYING FIGURE, and carry 2 to the product of the next figure, saying 3 times 4 is 12, and 2 we had to carry is 14. It now appears, that the product by the 3 tens is the same as that by the 3 units; but there is this difference-the figures in the product by the 3 tens are all removed one place further to the left, by which their value is increased TEN TIMES. (See Art. 13.) That this product should be ten times that by the 3 units, is obvious, because it is produced by a multiplier ten times as large (Art. 12.) And it is also obvious, that the sum of the products, by the 3 tens, and the 3 units, is equal to the product by 33. Hence, 35. To multiply simple numbers, (When the multiplier is more than 12,) RULE I. Multiply the multiplicand by EACH FIGURE in the multiplier and set down the first figure of each product directly under the MULTIPLYING FIgure. II. Add the several products together, and it will give the ANSWER. Illus. What will 57 head of cattle come to at 34 dollars a head? 571 It is immaterial, as to the result, which of the factors we multiply by; for 34 7 x 44 x 7: It is more convenient, however, to make the lesser factor the multipher. 228 171 N. B. Either the multiplicand or the multiplier is always to be considered as an abstract number (Art. 3, Note.) For, in strictness of language, it would be improper to speak of multiplying yards by dollars, or dollars by yards. These, and such like expressions may not, however, be wholly improper, if 1939 they be understood, not literally, but merely as a kind of convenient phrase, to indicate that the yards are to be multiplied by a NUMBER equal to the number of dollars, or that the dollars are to be multiplied by a NUMBER equal to the number of yards. 36. PROOF I. Cast the 9's out of the multiplicand and set the EXCESS on the left hand side of an Ex. II. Do the same with the multiplier, and set the excess on the right hand side. III. Multiply these two excesses together, cast the 9's out of their product, and set the excess above the EX. IV. Cast the 9's out of the total product and set the excess below; then if the excess ABOVE, be equal to the excess EELOW, the work may be supposed right. Illus. In 38 bushels of apples, each bushel containing 87 apples, how many apples? 87=6 3 Ans. 3306 = 3 I. Multiplicand. 8 and 7 is 15, 9's out leaves 6; which set on the left hand side. II. Multiplier. 3 and 8 is 11, 9's out leaves 2; which set on the right hand side. III. "Multiply these two excesses together;" 6 x 2 = 12, 9's out leaves 3; which set above the EX. IV. Product. 3 and 3 is 6, and 6 is 12; 9's out leaves 3; which set below the EX. NOTE. In adding together the digits, as above, the 9's need not be reckoned This method of proof depends upon a property of the number 9, which is, that" any number divided by 9, will leave the same remainder as the sum of its digits divided by 9." Illus. Take the number 465. This separated into its parts, becomes 400+60 +5. But 400= 4 × 99+ 4; and 60 = 6 x 9+ 6. Therefore 465 (4 X 99+4) (6x9+6)+5= 4 × 99+6x9 +4 + 6 + 5, and 465 divided by 9 = (4 x 37. When ciphers occur between any of the significant figures, NEGLEOT those ciphers, and multiply by the significant figures only. REMEMBER ALWAYS to place "the first figure of each product directly under the multiplying figure." 9. Multiply nine thousand eight hundred and seventy-six by five thou sand and nine. Ans. 49468884. 10. What will 354 yards of cloth come to at 107 cents per yard? Ans. 37878 cents. 11. Multiply eighty-nine thousand five hundred and six by three thousand and fifty-two. Ans. 273172312. 12. Bought 405 tons of hay for 984 cents per ton; what did it come to Ans. 398520 cents. CONTRACTIONS IN MULTIPLICATION. 38. A COMPOSITE NUMBER is a number composed of two or more other numbers MULTIPLIED TOGETHER. And 22, 36, 144, &c. are composite numbers. Because 22 11X2, 36=6×6, 144 12x12, or 8×6×3, &c. 39. The PARTS of which a composite number is composed are called its COMPONENT PARTS. Thus 11 & 2 are the component parts of 22; because 11 x 2 = 22. And 6 & 6 (or 4 & 9, or 3 & 12,) are the component parts of 36; because 6 x 6 (or 4 × 9, or 3 × 12)= 36. And so on. 40. If we multiply any number by 11, and that product by 2, we shall evidently obtain a product equal to (2 times 11 is) 22 times that number. If there should be three or more component parts, it is plain, that we first multiply by one of them, that product by another, this last product by a third, and so on. For instance, to multiply by 144 (the component parts of which being 8, 6 & 3,) we multiply first by 8, and obtain 8 times the given number; that product by 6, and obtain (6 times 8 is) 48 times the given number; and this last product by 3, and obtain (3 times 48 is) 144 times the given number. This method of proceeding may be applied to any composite number whatever. Therefore, To multiply by a composite number, RULE. Multiply by its COMPONENT PARTS continually. 99+6x9 +4 +6+5) ÷ 9. But 4 x 99 +6 x 9 is evidently divisible by 9 without a remainder; therefore the given number 465 divided by 9 will leave the same remainder as the sum of its digits divided by 9. * EXAMPLES. 1. How much will 39 yards of cotton cloth come to at 18 cents a yard? Ans. 702 cents. 2. Suppose there are 376 pensioners in one state, and each one receives 96 dollars per annum ;* how much will they all receive? (12 × 8=96.) Ans. 36096 dollars. 3. Suppose a man sows 13728 grains of wheat, and each grain produces 42 other grains; how many grains will he harvest? Ans. 576576. 41. Annexing a cipher to the right hand of any number removes it one place towards the left, and therefore increases its value TEN TIMES: and it is therefore, in effect, multiplying it by TEN. Hence as many ciphers as we annex to the right hand of a number, so many times we multiply that number by TEN. Therefore, When the multiplier is 1, with any number of ciphers annexed, RULE. Annex as many ciphers to the right hand of the MUL And so of any other number multiplied by 10, 100, 1000, &c. 42. From the principles already explained, it is obvious, that, When there are ciphers on the left hand of EITHER or BOTH the factors We may obtain the product by observing the following RULE. NEGLECT THOSE CIPHERS, place the other figures under one another, and multiply them as before directed; and to their product annex as many ciphers as were neglected in BOTH 4. There are 160 square rods in an acre; how many square rods then in 570 acres? Ans. 91200, 5. Bought 130 acres of land at 40 dollars per acre; how much did it come to? Ans. 5200 dollars. 6. Seventy men paid a certain sum, whereof each man paid 1300 dollars; how much did they all pay? Ans. 91000 dollars. 7. There are 60 minutes in an hour, and 24 hours in a day; how many minutes then are there in 1 day? 1 week? 1 year? (allowing 7 days to a week, and 52 weeks to the year. Ans. to last, 524160. THEORETICAL QUESTIONS. What is simple multiplication? What the sign? How is it read? What TERMS are used in multiplication? What is the multiplicand? -the multipliproduct? What are BOTH the multiplicand and the multiplicr sometimes er ? *A Latin phrase signifying by the year. |