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 units, 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 thousand?

dollars per

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 dollars per ton?

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.

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.

2. The process of finding the factors of which a given number is composed, is called resolving it into factors.

56. Some numbers may be resolved into more than two factors; and also into different sets of factors. Thus the factors of 24 are 3, 2, 2 and 2; or 4, 3 and 2; or 6, 2 and 2, or 8 and 3, or 6 and 4, or 12 and 2.

OBS. We have seen that the product of any two numbers is the same, whichever factor is taken for the multiplier. (Art. 47.) In like manner, the product of any three or more factors will be the same, in whatever order they are multiplied. For, the product of two factors may be considered as one number, and this may be taken either for the multiplicand or the multiplier. Again, the product of three factors may be considered as one number, and be taken for the multiplicand or the multiplier, &c. Thus 24-3×2×2×2=6×2× 2=12X2=6X4=4X2X3=8X3.

3. What will 24 hogsheads of molasses cost, at 37 dolapiece? Ans. 888 dollars. Suggestion. Resolve 24 into any two or more factors and proceed as before. Hence,

57. To multiply by a composite number.

Resolve the multiplier into two or more factors; multiply the multiplicand by one of these factors, and this product by another factor, and so on till you have multiplied by all the factors. The last product will be the product required.

OBS. The factors into which a number may be resolved, must not be confounded with the parts into which it may be separated. (Art. 26.) The former have reference to multiplication, the latter to addition; that is, factors must be multiplied together, but parts must be added together to produce the given number. Thus 56 may be resolved into two factors, 8 and 7; it may be separated into two parts, 5 tens or 50, and 6. Now 8X7-56, and 50+6=56.

4. What will 36 cows cost, at 19 dollars a head?

QUEST.-Obs. What is a composite number? What are the factors which produce it, sometimes called? What is meant by resolving a number into factors? 56. Are numbers ever composed of more than two factors? What are the factors of 24? 32? 36? 40? 42? 60? 64? 72? 108? Obs. When three or more factors are to be multiplied together, does it make any difference in what order they are taken? 57. When the multiplier is a composite number, how do you proceed? Obs. What is the difference between the factors into which a number may be resolved, and the parts into which it may be separated?

5. What cost 45 acres of land, at 110 dollars per acre? 6. At 36 shillings per week, how much will it cost a person to board 52 weeks?

7. If a man travels at the rate of 42 miles a day, how far will he travel in 205 days?

8. At the rate of 56 bushels per acre, how much corn can be raised on 460 acres of land?

9. What cost 672 yards of broadcloth, at 24 shillings per yard?

10. What cost 1265 yoke of oxen, at 72 dollars per yoke?

CASE II.-When the multiplier is 1 with ciphers annexed to it.

58. It is a fundamental principle of notation, that each removal of a figure one place towards the left, increases its value ten times; (Art. 9;) consequently, annexing a cipher to a number will increase its value ten times, or multiply it by 10; annexing two ciphers will increase its value a hundred times, or multiply it by 100; annexing three ciphers will increase it a thousand times, or multiply it by 1000, &c.; for each cipher annexed, removes each figure in the number one place towards the left. Thus 12 with a cipher annexed, becomes 120, and is the same as 12 × 10; 12 with two ciphers annexed, becomes 1200, and is the same 12 × 100; 12 with three ciphers annexed, becomes 12000, and is the same as 12 x 1000, &c. Hence,

59. To multiply by 10, 100, 1000, &c.

as

Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus formed will be the product required.

OBS. To annex means to place after or at the right hand.

11. What will 10 drums of figs weigh, at 28 pounds a drum? Ans. 280 pounds.

QUEST.-58. What effect does it have to remove a figure one place towards the left hand? Two places? 59. How do you proceed when the multiplier is 10, 100, 1000? Obs. What is the meaning of the term an