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is the same thing, (Art. 47,) tens into units, produces tens: consequently the 6 must be written in tens' place in the product; that is, under the figure 2 by which we are multiplying. 20 times 1 ten are 20 tens or 200; or simply say, 2 times 1 are 2: and since the 2 denotes hundreds, as we have just seen, set it on the left of the 6 in hundreds' place. 20 times 1 hundred are 20 hundred or 2000; or simply say, 2 times 1 are 2: and since the 2 denotes thousands, set it in thousands' place on the left of the last figure in the product. Finally, adding these two results together as they stand, units to units, tens to tens, &c., we have 2938 dollars, which is the whole product required.

OBS. The several products of the multiplicand into the separate figures of the multiplier, are called partial products. Hence,

52. When the multiplier contains more than one figure.

Multiply each figure of the multiplicand by each figure of the multiplier separately, placing each partial product in a separate line with the first figure of each line directly under that by which you multiply; the sum of the partial products will be the true product or answer required.

53. PROOF.-Multiply the multiplier by the multiplicand, and if the product thus obtained is the same as the other product, the work is supposed to be right.

OBS. 1. This method of proof depends upon the principle that the product of any two numbers is the same, whichever is taken for the multiplier. (Art. 47.)

2. When the multiplier is small, we may add the multiplicand to itself as many times as there are units in the multiplier, and if the sum is equal to the product, the work is right. Thus 78X3=234. Proof. 78+78+78=234, which is the same as the product.

3. Multiplication may also be proved by division and by casting out the nines; but neither of these methods can be explained here

QUEST.-Obs. What is meant by partial products? 52. How do you proceed when the multiplier contains more than one figure? How should the partial products be written? Where write the first figure of each line? What do you finally do with the partial products? 53. How is multiplication proved? Obs. On what principle does this method of proof depend? When the multiplier is small, how may we prove it?

without anticipating principles belonging to division, with which the learner is supposed as yet to be unacquainted.

22. What will 45 cows cost, at 27 dollars a head?

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23. What cost 63 hats, at 36 shillings apiece?

24. How much corn can a man raise on 87 acres, at 45 bushels per acre?

25. How many pounds of sugar will 75 boxes contain, if each box holds 256 pounds?

26. What cost 278 hogsheads of molasses, at 23 dollars per hogshead?

27. What is the product of 347 multiplied by 256?

Suggestion. Proceed in the same manner as when the multiplier contains but two figures, remembering to place the right hand figure of each partial product directly under the figure by which you multiply.

28. What is the product of 569 into 308? After multiplying by the 8 units, we must next multiply by the 3 hundreds, since there are no tens in the multiplier, and place the first figure of this partial product directly under the figure 3 by which we are multiplying.

Operation.

347

256

2082

1735

694

88832 Ans.

Operation.

569

308

4552

1707

175252 Ans.

29. What is the product of 67025 into 4005? Ans. 268435125.

30. What is the product of 841072 into 603 ?

54. From the preceding principles we derive the following

GENERAL RULE FOR MULTIPLICATION.

I. Write the multiplier under the multiplicand, units under units, tens under tens, &c.

II. When the multiplier contains but one figure,

Begin with the umts, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (Art. 49.)

III. When the multiplicand contains more than one figure,

Multiply each figure of the multiplicand by each figure of the multiplier separately, beginning at the right hand, and write the partial products in separate lines, placing the first figure of each line directly under the figure by which you multiply. (Art. 52.)

Finally, add the several partial products together, and the sum will be the whole product.

OBS. It is immaterial as to the result which of the factors is taken for the multiplier. (Art. 47). But it is more convenient and therefore customary to place the larger for the multiplicand and the smaller for the multiplier. Thus it is easier to multiply 254672381 by 7, than it is to multiply 7 by 254672381, but the product will be the same.

EXAMPLES FOR PRACTICE.

1. What will 465 hats cost, at 6 dollars apiece? 2. What will 638 sheep cost, at 4 dollars a head? 3. What will 1360 yards of cloth cost, at 7 dollars a yard?

4. What cost 169 bushels of potatoes, at 4 shillings per bushel?

5. What cost 279 barrels of salt, at 9 shillings a barrel? 6. At 12 dollars a suit, how much will it cost to furnish 1161 soldiers with a suit of clothes apiece?

7. What cost 1565 acres of wild land, at 7 dollars per acre?

QUEST.-54. What is the general rule for multiplication? Obs. Which number is usually taken for the multiplicand?

8. What will 758 baskets of peaches cost, at 5 dollars per basket?

9. What cost 25650 pounds of opium, at 6 dollars a pound?

10. How much can a man earn in 12 months, at 15 dollars per month?

11. What will 23 loads of hay come to, at 18 dollars a load?

12. What will 45 cows come to, at 21 dollars apiece? 13. What will 56 hogsheads of molasses cost, at 32 dollars a hogshead?

14. What cost 128 firkins of butter, at 13 dollars a firkin?

15. What cost 97 kegs of tobacco, at 26 dollars per keg? 16. What cost 110 barrels of pork, at 19 dollars per barrel ?

17. How much will 235 sheep come to, at 21 shillings a head?

18. How many bushels of corn will grow on 83 acres, at the average rate of 37 bushels to an acre?

19. In one bushel there are 32 quarts: how many quarts are there in 92 bushels ?

20. What will a drove of 463 cattle come to, at 48 dollars per head?

21. How much will 78 thousand of boards cost, at 19 dollars per thousand?

22. What cost 243 chests of tea, at 37 dollars per chest?

23. A man bought 168 horses, at 63 dollars apiece : what did they come to?

24. What cost 256 barrels of beef, at 16 dollars a barrel?

25. If 376 men can build a fortification in 95 days, how long would it take 1 man to build it?

26. Allowing 365 days to a year, how many days has a man lived who is 45 years old?

27. If a garrison consume 725 pounds of beef in one day, how many pounds will they consume in 125 days? 28. How many pounds will the same garrison consume in 243 days?

29. How far will a ship sail in 365 days, at 215 miles per day?

30. What cost 678 tons of Railroad iron, at 115 dolper ton?

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CONTRACTIONS IN MULTIPLICATION.

55. The general rule is adequate to the solution of all examples that occur in multiplication. In many instances, however, by the exercise of judgment in applying the preceding principles, the operation may be very much abridged.

CASE I.--When the multiplier is a composite number. Ex. 1. What will 14 hats cost, at 8 dollars apiece?

Suggestion. Since 14 is twice as much as 7; that is, 14-7x2, it is manifest that 14 hats will cost twice as much as 7 hats.

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we first multiply by the factor 7, and that product by 2, the other factor of 14.

Proof.

14

8

112 the same as before.

2. What will 27 horses cost, at 85 dollars apiece? Suggestion. Find the factors of 27; that is, find two numbers, which being multiplied together, produce 27, and multiply first by one of these factors and the product thus arising by the other.

OBS. 1. Any number which is the product of two or more factors, is called a composite number; and the factors, which being multiplied together, produce the composite number, are sometimes called the component parts of the number. Thus 14, 27, 32, &c., are composite numbers, and the factors 7 and 2, 9 and 3, 8 and 4, are their component parts.

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