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7. A man who has in his possession $6978, owes $1364 to one man, $965 to another, $1167 to another, and $847 to another. How many dollars will he have left after paying his debts ?

8. I bought a carriage for $147.13, and paid $11.48 for having it repaired, and $13.27 for having it painted. I let it enough to come to $19.75, and then sold it for $175. What did I gain by the transaction ?

9. A London merchant bought some merchandise for £247 88. 11d. For how much must he sell it, to gain £89 15s. 70.?

10. I bought a lot of flour for $912.46, and sold it for $1258.93. I afterwards bought a lot of wheat for $1678.27. For how much must I sell the wheat, to gain as much as I gained on the flour?

11. The distance from Albany to Buffalo, by railroad, is 325 miles. If A should start from Albany, and B should at the same time start from Buffalo, and travel towards each other, how far apart will they be when A has travelled 97 miles, and B has travelled 113 miles ?

12. A man started from Boston on a journey, taking with him $100. He took the cars for Providence, paying $1.50 for his ticket. He stayed in Providence 2 days, at an expense of $5.37, and then purchased a ticket for New York, for $3. He remained in New York 5 days, at an expense of $15.62, and then went to Philadelphia, paying $3 for his ticket, and $1.25 for hack hire. His expenses at Philadelphia were $20.75. The total expense of his homeward trip was $15.27. How much money had he left?

13. How much more money must a person get who has $1378.56, in order to purchase a house worth $2538, and still have $893 left with which to purchase furniture ?

14. I bought a wood-lot for $639.46, from which I obtained 123 ) cd. 5 cd. ft. of wood, worth $370.50; 75 cd. 6 cd. ft. worth $189.375; 84 cd. 4 cd. ft. worth $253.50; and 87 cd. 2 cd. ft. worth $174.50. How much wood did I obtain, in all? How many dollars was it worth ?

15. Allowing that it cost me $178.25 to have the wood named in the preceding problem cut; that my other expenses on account of it amounted to $48.27; and that, after cutting the wood, I sold the land for $113, did I gain or lose by the whole transaction, and how much?

SECTION VIII.

MULTIPLICATION.

47. Definitions and Explanations.

(a.) MULTIPLICATION is a process by which we ascertain how many units there are in any number of times a given number.

(b.) The number to be taken is called the MULTIPLICAND; the number showing how many times it is to be taken is called the MULTIPLIER ; and the result is called the PRODUCT.

(c.) The multiplier and multiplicand are called FACTORS of the product. The product is called a MULTIPLE of its factors.

ILLUSTRATION. - In 17 times 8 = 56,” or, “8 multiplied by 7 = 56," 8 is the multiplicand, 7 is the multiplier, and 56 is the product. 7 and 8 are factors of 56, and 56 is a multiple of 7 and 8.

(d.) The following tables should be thoroughly mastered. 2 times 1, or once 2 = 2.

3 times 3

= 9. 2 times 2

= 4.

3 times 4, or 4 times 3 = 12. 2 times 3, or 3 times 2 = 6.

3 times 5, or 5 times 3 = 15. 2 times 4, or 4 times 2

3 times 6, or 6 times 8 = 18. 2 times 5, or 5 times 2

3 times 7, or 7 times 3 = 21. 2 times 6, or 6 times 2

3 times 8, or 8 times 3 = 24. 2 times 7, or 7 times 2

3 times 9, or 9 times 3 = 27. 2 times 8, or 8 times 2 =

3 times 10, or 10 times 3 = 30. 2 times 9, or 9 times 2 = 2 times 10, or 10 times 2 = 20.

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8 times 8

= 64.
9 times 9

= 81. 8 times 9, or 9 times 8 = 72.

9 times 10, or 10 times 9 = 90. 8 times 10, or 10 times 8 = 80.

10 times 10 = 100, (e.) The above table shows that a change in the order of factors does not affect their product.

NOTE. — When the multiplicand is a concrete number, the principle is applied thus; 8 times 6 books = 6 times 8 books; 9 times 7 cents = 7 times 9 cents.

(f.) The multiplier must always be an abstract number.

ILLUSTRATIONS.— We can multiply a number by 5 or 7, i. e. we can take it 5 times or 7 times; but it would be absurd to speak of multiplying it by 5 cents or 7 bushels, i. e. of taking it 5 cents or 7 bushels times.

(g.) The product is always of the same denomination as the multiplicand.

ILLUSTRATIONS.- 9 times 8 = 72; 9 times .08 = .72; 9 times 8 bushels = 72 bushels.

(h.) In multiplying, we consider each denomination separately, beginning at the right. We reduce as much of each product as possible to units of the next higher denomination, writing the remainder, and adding the reduced units to the product of the next denomination.

1. What is the product of 86.3 multiplied by 7? 86.3 Multiplicand. SOLUTION.— Writing the numbers as opposite, 7 Multiplier.

we multiply thus:

7 times 3 tenths = 21 tenths = 2 units and 1 604.1 Product.

tenth. Writing the 1 as the tenths figure of the product, we add the 2 units to the product of the units, thus :

7 times 6 units = 42 units, and 2 units are 44 units = 4 tens and 4 units. Writing 4 as the units figure of the product, we add the 4 tens to the pro. duct of the tens, thus:

7 times 8 tens = 56 tens, and 4 tens = 60 tens, which, being the last denomination, we write.

The answer, then, is 604.1.

(i.) Methods of Proof. — 1st. Perform the work again carefully, as before.

2d. Change the order of the factors.

3d. Separate the multiplier into two or more parts, multiply by the parts separately, and find the sum of the products, thus; 7 times 86.3 = 4 times 86.3 + 3 times 86.3.

NOTE. — Each result should equal that first obtained; otherwise, there is an error, which should be discovered and corrected.

4th. Those who understand Division, may prove the work by dividing the product by one of the factors. The quotient should equal the other factor.

48. Examples and Practical Problems. (a.) How many are 1. 3 times 8543 ?

16. 9 times 987654? 2. 9 times 2435 ?

17. 2 times 135796 ? 3. 4 times 9378 ?

18. 5 times .0075937 ? 4. 6 times 58794?

19. 3 times 867.149 ? 5. 2 times 930.67 ?

20. 8 times .008672 ? 6. 5 times 279.48 ?

21. 6 times 400.597 ? 7. 8 times 6583.2?

22. 7 times 93.1425 ? 8. 7 times 400.674?

23. 4 times 796896 ? 9. 4 times 909.99 ?

24. 8 times 93472 ? 10. 5 times 376.804 ?

25. 8 times 8976 ? 11. 9 times .07086 ?

26. 9 times 10438 ? 12. 3 times 777.898?

27. 5 times 18.198. 13. 7 times 1234321 ?

28. 9 times .07658 ? 14. 6 times 875908 ?

29. 4 times 43.019? 15. 4 times 96.369 ?

30. 8 times 7958.99 ? 31. What will 7 acres of land cost at $428.36 per acre ?

REASONING PROCESS. - If 1 acre of land costs $428.36, 7 acres will cost 7 times $428.36, which may be found by multiplying by 7.

(b.) How much will —
32. 9 barrels of beef cost at $23.125 per bbl. ?
33. 7 quintals of fish cost at $3.75 per quintal ?
34. 6 bales of cotton weigh if each weighs 296 lb. ?

35. 4 acres of land cost at $245.70 per acre ?
36. 7 barrels of flour weigh if each weighs 196 lb. ?
37. 5 lots of land contain if each contains 3.795 acres ?
38. 4 stacks of hay weigh if each weighs 6478 lb. ?
39. A man earn in 6 months if he earns $37.75 per month ?
40. 2 lots of land cost at $1397 each ?
41. 247 barrels of flour cost at $8 per bbl. ?

REASONING PROCESS. - If one bbl. of flour costs $8, 247 bbl. will cost 247 times $8, which is equivalent to 8 times $247. Hence, the answer may be found by multiplying $247 by 8.

Note. - The change in the order of the factors is made because the multiplier is larger than the multiplicand.

How much will —
42. 5794 soldiers earn in 1 month if each earns $8?
43. 1147 bottles hold if each holds 3 pints ?
44. 6978 ears of corn contain if each ear contains 4 gi. ?
45. 1746 paces measure if each pace measures 3 feet?
46. 957 pounds of sugar cost at $.09 per lb. ?

REASONING PROCESS. - If 1 pound of sugar cost 9 cents, 957 lb. will cost 957 times 9 cents, which is equivalent to 9 times 957 cents, or to 9 times $9.57. Hence, the answer may be found by multiplying $9.57 by 9.

How much will —
47. 1849 pounds of iron cost at $.05 per lb. ?
48. 1296 paper boxes cost at $.08 each?
49. 1397 pounds of raisins cost at $.07 per lb. ?
60. 249 oranges cost at $.03 each?
51. 2050 bars of soap weigh if each bar weighs 2 lb.
52. 578 papers of pins cost at $.08 per paper ?
53. 1756 miles = how many furlongs ?

REASONING Process. — Since 1 mile equals 8 furlongs, 1756 miles must equal 1756 times 8 furlongs, equivalent to 8 times 1756 furlongs ?

54. 2785 gallons = how many quarts ?
55. 5947 square yards = how many square feet ?

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