Thus, when we say 6 times 12 are 72, 12 is the multiplicand, 6 the multiplier, and 72 the product. 44. The multiplier and multiplicand together are often called factors, because they make or produce the product. OBS. 1. The term factor is derived from a Latin word which signifies an agent, a doer, or producer. 2. When the multiplicand denotes things of one denomination only, the operation is called Simple Multiplication. 45. Multiplying by 1 is taking the multiplicand once: thus, 4 multiplied by 1=4. Multiplying by 2 is taking the multiplicand twice: thus, 2 times 4, or 4+4=8. Multiplying by 3 is taking the multiplicand three times : thus 3 times 4, or 4+4+4=12, &c. Hence, Multiplying by any whole number is taking the multiplicand as many times as there are units in the multiplier. Note. The application of this principle to fractional multipliers will be illustrated under fractions. OBS. 1. From the definition of multiplication, it is manifest that the product is of the same kind or denomination as the multiplicand: for repeating a number or quantity does not alter its nature. Thus, if the multiplicand is an abstract number; that is, a number which does not express money, yards, pounds, bushels, or have reference to any particular object, the product will be an abstract number; if the multiplicand is money, the product will be money; if weight, the product will be weight; if measure, measure, &c. 2. Every multiplier is to be considered an abstract number. In familiar language it is sometimes said, that the price multiplied by the weight will give the value of an article, and it is often asked how much 5 cents multiplied by 5 cents, &c., will produce. But these are abbreviated expressions, and are liable to convey an erroneous idea, or rather no idea at all. If taken literally, they are absurd; for multiplication is repeating a number or quantity a certain number of times. Now to say that the price is repeated as many times as the given quantity is heavy, or that 5 cents are repeated 5 cents times, is nonsense. But we can multiply the price of 1 pound by a number equal to the number of pounds in the weight of the given article, and QUEST.-When we say, 6 times 9 are 54, what is the 6 called? The 9? The 54? 44. What are the multiplicand and multiplier together called? Why? Obs. What does the term factor signify? 45. What is meant by multiplying by 1? By 2? By 3? What is it to multiply by any whole number? the product will be the value of the article. We can also multiply 5 cents by the number 5; that is, repeat 5 cents 5 times, and the product is 25 cents. Construed in this manner, the multiplier becomes an abstract number, and the expressions have a consistent meaning. 46. Multiplication is often denoted by two oblique lines crossing each other ×, called the sign of multiplication. It shows that the numbers between which it is placed, are to be multiplied together. Thus the expression 9×6, signifies that 9 and 6 are to be multiplied together, and is read, "9 multiplied by 6," or, simply, "9 into 6." OBS. The product will be the same, whether we multiply 9 by 6, or 6 by 9; for, by the table, 6 times 9 are 54, also 9 times 6 are 54. So 6X4 4×6; 5×3=3×5; 8x7=7x8, &c. *** *** To illustrate this point; suppose there is a certain orchard which contains 4 rows of trees, and each row has 6 trees. Let the number of rows be represented by the number of horizontal rows of stars in the margin, and the number of trees in each row by the number of stars in a row. Now it is evident that the whole number of trees in the orchard is equal either to the number of stars in a horizontal row repeated four times, or to the number of stars in a perpendicular row repeated six times; that is, equal to 6×4, or 4x6. Hence, *** 47. The product of any two numbers will be the same, whichever factor is taken for the multiplier. EXERCISES FOR THE SLATE. Ex. 1. What will 3 house-lots cost, at 231 dollars each? Suggestion. If 1 house-lot costs 231 dollars, 3 lots will cost 3 times 231 dollars; that is, three lots will cost 231+231+231, or 693 dollars. QUEST. Obs. Of what denomination is the product? How does this appear? What must every multiplier be considered? Can you multiply by a given weight, a measure, or a sum of money? 46. How is multiplication sometimes denoted? What does the sign of multiplication show? How is the expression 9 x 6 read? How 6x7=42? 47. Does it make any difference in the product, which factor is made the multiplier? How illustrate this? Having written the numbers upon the slate, as in the margin, proceed thus: 3 times 1 unit are 3 units. Set the 3 in units' place under the multi Operation. 231 Multiplicand. 3 Multiplier. Dolls. 693 Product. plier. 3 times 3 tens are 9 tens; set the 9 in tens' place. 3 times 2 hundreds are 6 hundreds; set the 6 in hundreds' place. The product is 693 dollars. 2. What will 4 horses cost, at 120 dollars apiece? Suggestion. Write the numbers, &c., and proceed as before. Ans. 480 dolls. 3. What is the product of 312 multiplied by 3? Ans. 936. 4. What is the product of 121 multiplied by 4? Ans. 484. rods are 5. In 1 mile there are 320 rods: how many there in 3 miles? 6. If a man travel 110 miles in 1 day, how far can he travel in 8 days? 11. What will 6 stage-coaches cost, at 783 dollars apiece? Operation. 783 6 Proceeding as before, 6 times 3 units are 18 units, or simply say, 6 times 3 are 18. Now 18 requires two figures to express it; hence, we set the 8 under the figure multiplied, and Ans. 4698 dolls. reserving the 1, carry it to the next product, as in addition. (Art. 25.) 6 times 8 are 48, and 1 (to carry) makes 49. Set the 9 under the figure multiplied, and carry the 4 to the next product, as before. 6 times 7 are 42, and 4 (to carry) make 46. Since there are no more figures to be multiplied, set down the 46 in full. The product is 4698 dollars. Hence, 49. When the multiplier contains but one figure. Write the multiplier under the multiplicand; then, beginning at the right hand, multiply each figure of the multiplicand by the multiplier separately; if the product of any figure of the multiplicand into the multiplier does not exceed 9, set it in its proper place under the multiplier, but if it does exceed 9, write the units' figure under the figure multiplied, and carry the tens to the next product on the left, as in addition. (Art. 25.) 50. The principle of carrying the tens in multiplication is the same as in addition, and may be illustrated in a similar manner. (Art. 26.) Take, for instance, the last example, and set the product of each figure in a separate line. In this analytic process it will be seen that the tens' figure of each product which exceeds 9, is added to the next product on the left, the same as in the common operation above The only difference between the two operations is, that in one we add the tens as we proceed in the multiplication; in the other, we reserve them till each figure is multiplied, and then add them to the same orders as before consequently, the result must be the same in both. (Art. 27.) QUEST.-49. How do you write the numbers for multiplication? Where begin to multiply? When the product of a figure in the multiplicand does not exceed 9, where is it written? When it exceeds 9, what is to be done with it? 50. How is the principle of carrying in multiplication illustrated? 51. From this and the preceding illustrations, the learner will perceive that units multiplied by units produce units; tens into units produce tens; hundreds into units produce hundreds, &c. Hence, When the multiplier is units, the product will always be of the same order as the figure multiplied. 12. What cost 83 pounds of opium, at 8 dollars per pound? 13. At 9 shillings per day, how much can a man earn in 213 days? 14. If 1 sofa costs 78 dollars, how much will 8 sofas cost? 15. What cost 879 barrels of flour, at 7 dollars a barrel ? 16. At 8 shillings apiece, what will a drove of 650 lambs come to ? 21. What will 26 horses cost, at 113 dollars a piece? Suggestion. Reasoning as before, if 1 horse cost 113 dollars, 26 horses will cost 26 times as much. Operation. 678 cost of 6 horses. 226* cost of 20 66 Since it is not convenient to multiply by 26 at once, we first multiply by the 6 units, then by the 2 tens and add the two results together.Thus 6 times 3 are 18; set down the 8 and carry the 1, as above. are 6, and 1 to carry makes 7. 6 times 1 are 6. Next multiply by the 2 tens thus: 20 times 3 units are 60 units or 6 tens; or we may simply say, 2 times 3 are 6: Now the 6 must denote tens; for units into tens, or what Ans. 2938 cost of 26 6 times 1 QUEST.-51. What do units multiplied into units produce? Tens into units? Of what order is the product universally when the multiplier is units? |