(2.) The results of multiplying by 5 terminate in 5 and 0, alternately. Thus, 5 times 1 are 5; 5 times 2 are 10; 5 times 3 are 15, &c. (3.) The first nine results of multiplying by 11 are formed by repeating the figure to be multiplied. Thus, 11 times 2 are 22; 11 times 3 are 33, &c. (4.) In the successive results of multiplying by 9, the right hand figure regularly decreases by 1, and the left hand figure regularly increases by 1. Thus, 9 times 2 are 18; 9 times 3 are 27; 9 times 4 are 36, &c. 13. At 2 dollars a cord, what will 12 cords of wood cost? 10 cords? 9 cords? 8 cords? 7 cords? 6 cords? 5 cords? 4 cords? 3 cords? 14. In one yard there are 3 feet: how many feet are there in 12 yards? in 11 yards? 10 yards? 9 yards? 8 yards? 7 yards? 6 yards? 5 yards? 4 yards? 15. In one gallon there are 4 quarts: how many quarts in 12 gallons? in 11 gallons? 10 gallons? 9 gallons? 8 gallons? 7 gallons? 6 gallons? 5 gallons? 4 gallons? 16. If you buy 5 marbles for a cent, how many can you buy for 12 cents? for 11 cents? 10 cents? 9 cents? 8 cents? 7 cents? 6 cents? 5 cents? 4 cents? 17. In New England, a dollar contains 6 shillings; how many shillings do 12 dollars contain? 11 dolls.? 10 dolls.? 9 dolls. ? 8 dolls.? 7 dolls.? 6 dolls.? 5 dolls.? 18. If 7 pounds of sugar cost a dollar, how many pounds can you buy for 12 dollars? for 11 dollars? 10 dolls? 9 dolls. ? 8 dolls. ? 7 dolls. ? 6 dolls.? 5 dolls. ? 19. In New York, a dollar contains 8 shillings: how many shillings do 12 dollars contain? 11 dolls.? 10 dolls. ? 9 dolls.? 8 dolls? 7 dolls.? 6 dolls.? 5 dolls. ? 20. At 9 cents a quart, what will 12 quarts of blackberries cost? 11 quarts? 10 quarts? 9 quarts? 8 quarts? 7 quarts? 6 quarts? 5 quarts? 21. What will be the cost of 12 yards of silk, at 10 shillings per yard? of 11 yards? 10 yards? 9 yards? 8 yards? 7 yards? 6 yards? 5 yards? 22. What cost 8 cords of wood, at 5 dollars per cord? 23. If 7 yards of cloth make a cloak, how many yards will it take to make 8 cloaks? 24. What is the cost of 9 pounds of ginger, at 8 cents a pound? 25. At 12 dollars apiece, what will 10 cows cost? 26. What cost 10 barrels of cider, at 9 shillings a barrel? 27. What will 11 pair of shoes come to, at 10 shillings a pair? 28. If 8 men can do a piece of work in 9 days, how long will it take 1 man to do it? 29. If a barrel of beer will last 7 persons 8 weeks, how long will it last 1 person? 30. How much will 3 cows cost, at 14 dollars apiece? Suggestion. 14 is composed of 1 ten and 4 units, or 10 and 4. Now, 3 times 10 are 30, and 3 times 4 are 12; but 12 added to 30 make 42. Hence, 3 times 14 dollars are 42 dollars. Ans. 42 dollars. 31. What will 5 tons of hay cost, at 13 dollars per ton ? 32. What cost 4 hogsheads of molasses, at 15 dollars per hogshead? 33 How much can a man earn in 6 months, at 15 dollars per month? 34. A butcher bought 6 sheep, at 17 shillings apiece : how many shillings did they come to? 35. If a scholar performs 18 examples in 1 day, how many can he perform in 5 days? 36. In 1 pound there are 16 ounces: how many ounces are there in 8 pounds? 37. How far will a man walk in 5 days, if he walks 20 miles per day? 38. If 19 men can build a house in 4 days, how long would it take one man to do it? 39. If a shoemaker packs 16 pair of boots in 1 box, how many pair can he pack in 7 boxes? 40. If 1 acre of land produces 23 bushels of wheat, how many bushels will 4 acres produce? Suggestion. 23 is composed of 2 tens and 3 units, or 20 and 3. Now, 4 times 20 are 80; 4 times 3 are 12: 12 and 80 are 92. Ans. 92 bushels. 41 A merchant bought 4 pieces of silk, each piece having 24 yards: how many yards did they all contain? 42. What will 6 sleighs cost, at 25 dollars apiece? 43. What cost 4 reading books, at 42 cents apiece? 44. In 1 guinea there are 21 shillings: how many shillings are there in 5 guineas? 45. In one hogshead there are 63 gallons: how many gallons are there in 4 hogsheads? 46. What cost 32 pounds of sugar, at 8 cents per pound? 47. What cost 85 reams of paper, at 3 dollars per ream? 48. What cost 90 hats, at 4 dollars apiece? 49. In 1 week there are 7 days; how many days are there in 70 weeks? 50. In 1 hour there are 60 minutes: how many minutes are there in 9 hours? 42. Let us now attend to the nature of the preceding operations in this section. Take, for instance, the first example. Since 1 pencil costs 4 cents, 3 pencils will cost 3 times 4 (cents). Now 3 times 4 (cents) is the same as 4 (cents) + 4 (cents) + 4 (cents), viz: 12 cents. Again, in the second example: since 1 yard of cloth costs 6 dollars, 4 yards will cost 4 times 6 dollars and 4 times 6 dollars is the same as 6 (dollars) + 6 (dollars) +6 (dollars,) +6 (dollars), viz: 24 dollars. 43. This repeated addition of a number or quantity to itself, is called MULTIPLICATION. The number to be repeated, or multiplied, is called the multiplicand. The number by which we multiply, or which shows how many times the multiplicand is to be repeated, is called the multiplier. The number produced, or the answer, is called the product. QUEST. 43. What is multiplication? What is the number to be repeated called? What the number by which we multiply? What does the multiplier show? What is the number produced called? When we say, 6 times 12 are 72, which is the multiplicand? Which the multiplier? Which the product? 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 X, 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 4×6. 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? |