5. When wood is worth $6 a cord, what shall 1 pay for 3 cord£ ? 5 cords? 8 cords? 11 cords? 6. In one year there are 12 months, how many months in 2 years? 4 years? 7 years? 12 years? 7. If I study 11 hours in a day, how many hours shall I study in 3 days? 5 days? 8 days? 11 days? 56. To multiply by a single figure. 8. In one bushel are 32 quarts; how many quarts in 6 bushels? BY ADDITION. 6 BY MULTIPLICATION. 32 32 32 32 32 32 Product, 192 In 6 bushels there are, evidently, 6 times as many quarts as in 1 bushel, and the number of quarts in 6 bushels may be obtained by adding, as in the margin; or, more briefly, by multiplying; thus, 6 times 2 units are 12 units=1 ten and 2 units; write the 2′ units in units' place, and then say 6 times 3 tens are 18 tens, which, increased by the 1 ten previously obtained, make 19 tens 1 hundred and 9 tens, and these, written in the place of hundreds and tens respectively, give the true product. Hence, 32 Sum, 192 RULE. Write the multiplier under the multiplicand, and draw a line beneath; multiply the units of the multiplicand, set the units of the product under the multiplier, and add the tens, if any, to the product of the tens, and so proceed. 56. Which figure of the Multiplicand is multiplied first? Where are the units of the product written? What is done with the tens? Repeat the rule. 57. To multiply by two or more figures. 19. How many quarts in 46 bushels? OPERATION. Multiplicand, 32 192 128 First multiply by 6, as though 6 were the only figure in the multiplier; then multiply by 4, and set the first figure of this product in the place of tens; for multiplying by the 4 tens is the same as multiplying by 40, and 40 times 2 units are 80 units: 8 tens; i. e. the product of units by tens is tens. Having multiplied by each figure in the multiplier, the sum of the partial products will be the true product. Product, 1472 NOTE. So much of the product as is obtained by multiplying the whole multiplicand by one figure of the multiplier is called a partial product; thus, in the 19th example, 192 is the first partial product and 128 tens is the second. 58. Similar reasoning applies however many figures there may be in the multiplier. Hence, RULE. 1. Set the multiplier under the multiplicand and draw a line beneath. 2. Beginning at the right hand of the multiplicand, multiply the multiplicand by each figure in the multiplier, setting the first figure of each partial product directly under the figure of the multiplier which produces it. 3. The sum of these partial products will be the true product. 59. PROOF. Multiply the multiplier by the multiplicand, and, if correct, the result will be like the first product. NOTE. This proof rests on the principle, that the order of the factors is immaterial; thus, 3×4=4X3; 5×2×7=2X7X5. 57. Which figure of the multiplier is first employed? Where is the first figure of each partial product written? What is a partial product? 58. Rule for multiplying by two or more figures? 59. Proof? Principle? 60. The sign of multiplication, X, signifies that the two numbers between which it stands are to be multiplied together; thus, 6 X 530, i. e. six multiplied by five equals thirty; or, more familiarly, six times five are thirty. 31. How many are 726 × 27? 37. 4620 × 524 =? 39. 7692 X 356=? 40. 2146 X 179=? Ans. 19602. Ans. 2563912. Ans. 131328. Ans. 230850. 41. 37642 × 57 =? 789 = ? · 60. Sign of multiplication, what does it signify? 46. If 37 men do a piece of work in 23 days, in how many days will 1 man do the same work? 47. What is the value of 37 acres of land, at $43 per acre? 48. If a horse can travel 41 miles per day, how far can he travel in 17 days? 49. How many yards of cloth in 29 pieces, if each piece contains 31 yards? 61. To multiply by a composite number. A COMPOSITE NUMBER is the product of two or more numbers; thus 15 is a composite number, whose factors are 3 and 5; and 12 is a composite number, whose factors are 2 and 6, or 3 and 4, or 2, 2, and 3. It will be observed that a composite number may have several sets of factors. 50. If 35 men have $37 each, how many dollars have they all? 61. What is a composite number? May a composite number have more than one set of factors? Several other sets of factors of 168 may be used, and give the same product. Every similar example may be solved in like Hence, manner. RULE. Multiply the multiplicand by one of the factors of the multiplier, and that product by another factor, and so on until all the factors in the set have been taken; the last product will be the true product. 52. Multiply 743 by 42, i. e. by 7 and 6. 53. Multiply 3467 by 56. 54. 839 × 54 how many? = Ans. 31206. Ans. 45306. Ans. 827658432. 62. To multiply by 10, 100, 1000, or 1 with any nnmber of ciphers annexed: RULE. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number so formed will be the product. NOTE. The reason of the rule is obvious. Annexing a cipher removes each figure in the multiplicand one place toward the left, and thus its value is made ten fold (Art. 15). 60. Multiply 74 by 10. 61. Multiply 869 by 10000. 62. Multiply 4698 by 1000. 63. 76984 X 100000 =? 64. 59874 X 1000000000 =? Ans. 740. Ans. 8690000. Ans. 7698400000. 63. To multiply by 20, 50, 500, 25000, or any similar number: RULE. Multiply by the significant figures, and to the product annex as many ciphers as there are ciphers at the right of the significant figures of the multiplier. 61. Rule for multiplying by a composite number? 62. How is a number multipled by 10? By 100? Why? 63. How is a number multiplied by 20? Why? |