Hence, to 465 X 1, which is 465, we annex two ciphers, and have 100 times the number, or 46500, the answer required.

same rate, can you travel in 100 hours?

15. What will 45 casks of molasses cost at 50 dollars each? 16. If there are 640 acres in a square mile, how many acres are there in 150 square miles?

Ans. 96000.

75. Rule for Multiplication.-If the multiplier consists of one order of units, multiply each order of the multiplicand, beginning at the right, by the multiplier. Write the units of each result in the product, and reserve the tens, if any, to be added to the next result.

If the multiplier consists of more than one order of units, multiply each order of the multiplicand by each order of the multiplier, write the right-hand figure of each partial product under the order of the multiplier used, and add the partial products.

If either factor has one or more ciphers on the right, multiply without regard to these ciphers, and annex to the result as many ciphers as are on the right of both the factors.

PROOF.-Review the work, or reverse the order of the factors and multiply. If the work is correct, the result will be the same by both methods.

6. 555 by 4.

3. 1315 by 6. Ans. 7890. 14. 248 by 19. 4. 116 by 1000.

5. 413 by 70. Ans. 28910. 16. 365 by 37.

7. 4163 by 7. Ans. 29141. 18. 4561 by 603. Ans. 2750283.

Ans. 4712.

15. 160 by 30.

Ans. 13505.

17. 1040 by 11.

345000 = 345 × 1000

345 = 345 ×

1

344655 = 345 × 999

Ans. 1292928.

SOLUTION. Since 1000 times any number, less once the number, must be 999 times the number, we here abridge the solution by taking the multiplicand 1000 times, or once too many, by annexing three ciphers, and then subtracting the multiplier. This method of abridgment applies whenever the multiplier is 1 less than 100, 1000, 10000, etc.

25. 4573 X 99 26. 1316 × 999

what number?

what number?

Ans. 452727.

= what number?

Ans. 12298770.

28. What is the product of 1036 by 990?

27. 1230 × 9999

1

29. What is the product of 4455 by 105?

4455004455 × 100

222754455 X

5

467775

4455-X 105

SOLUTION.-Here the solution may be abridged by writing the product of 4455 by the 1 hundred at once, by annexing two ciphers, and writing

under it the product of 4455 by the 5 ones, and then adding the two partial products.

30. What is the product of 6307 by 1003?

31. If 5362 feet of boards can be sawed in a mill in one day, how many feet can be sawed in it in 313 days?

Ans. 1678306. 32. At 125 dollars a month, how many dollars can be earned in 12 months? Ans. 1500. 33. What will 3158 tons of coal cost at 8 dollars per ton?

3158

8 25264

SOLUTION.-At 8 dollars a ton, 3158 tons of coal will cost 3158 times 8 dollars, which is equal to 8 times 3158 dollars, or 25264 dollars.

8 dollars is the true multiplicand, but since 8 times 3158 gives the same product as 3158 times 8, we, for convenience, in the solution consider both factors as abstract numbers, and make the smaller factor the multiplier.

34. What will 8344 yards of cloth cost at 6 dollars a yard? Ans. 50064 dollars. 35. How many oranges in 47 boxes when each box contains 279 oranges? Ans. 13113. 36. Two factors are 7312 and 7000. What is their product? Ans. 51184000.

37. There were 6 drawers in a desk, 8 compartments in each drawer, and 87 dollars in each compartment. How many dollars did the desk contain?

38. What is the continued product of the factors 6, 8 and 87? 39. If 15 pounds of hay are required by 1 horse for 1 day, how many pounds are required by 5 horses for 30 days?

40. How many gallons in 1025 casks, if each cask contain 63 gallons? Ans. 64575. How many seeds Ans. 22209750.

41. In a bushel of rye are 888390 seeds. are there in 25 bushels?

42. If the weight of a cubic foot of white-oak wood is 43 pounds, what is the weight of 128 cubic feet?

43. What is the continued product of the factors 12, 420 and 310? Ans. 1562400.

44. How many yards of cloth in 32 bales, each bale having 121 pieces, and each piece 31 yards? Ans. 120032.

45. What is the population of a State containing 8320 square miles, if each square mile has 91 inhabitants?

46. If Pennsylvania contain 46000 square miles, what will be its population at 75 persons to a square mile?

Ans. 3450000 persons.

47. Sound moves 1142 feet in a second. How far will it move in one hour, or 3600 seconds? Ans. 4111200 feet. 48. If a railroad, 1035 miles in length, should obtain government aid to the amount of 52400 dollars per mile, what would be the entire amount received?

TEST QUESTIONS.

76.-1. What is MULTIPLICATION? The product? The factors of a product? The multiplicand? The multiplier? What does the sign of multiplication denote when written between two numbers?

2. What are the PRINCIPLES of multiplication? Show that the multiplicand and product are similar numbers. Why must the multiplier be regarded always as an abstract number? Show that the product is the same in whatever order its factors are used.

3. Recite the RULE for multiplication. Give the proof. Why, in multiplication, do you begin at the right to multiply? When the multiplier consists of more than one order of units, how many partial products may there be? What will be the unit of the partial product when the multiplier is a number of simple units? When the multiplier is a number of tens? When there are partial products, how do you obtain the entire product?

77.-Ex. 1. How many barrels, holding 3 bushels each, will be required to hold 30 bushels of apples?

2. If a farmer should raise 21 bushels of potatoes, and should wish to put them into barrels holding 3 bushels each, how many barrels would be required?

3. If you have 28 apples, and wish to distribute them equally among 7 boys, how many can you give to each boy? 4. How many times are 9 bushels contained in 36 bushels? What is the product of 9 bushels by 4?

5. How can you show that 9 bushels are contained in 36 bushels 4 times?

6. When you distribute 35 apples equally among 7 boys, do you find how many times 7 boys are contained in 35 apples, or do you find one of the 7 equal parts of 35?

7. When you find how many times 9 cents are contained in 72 cents, is the result a concrete or an abstract number?

8. When you find one of the 9 equal parts of 72 cents, is the result a concrete or an abstract number?

9. How many times are 9 dollars contained in 29 dollars, and how many dollars remain?