even hundreds will be dollars; and hence, if we cut off the two low. er orders, viz, tens and units, they will be cents, and all to the left will be dollars. According to this principle, the answer in the 5th example is 1 dollar 92 cents; and in the last, 66 dollars 24 cents.

12. A man bought 17 barrels of ale at 9 dollars a barrel; and 25 barrels of flour at 7 dollars a barrel; and 9 pieces of broadcloth, each piece containing 47 yards, at 6 dollars a yard. What did it all Ans. 2,866 dolls.

come to ?

13. A man bought 15 pounds of beef at 7 cents a pound; 8 pounds of sugar at 18 cents a pound; 9 yards of cotton cloth at 22 cents a yard; and 3 pails at 62 cents apiece. What did the whole amount to? Ans. 6 dolls. 33 cts. Ans. 1,095 | 19. 74,635X3 Ans. 223,905

14, 365X3

2,184 20. 53,894×4

2,358 21. 83,528X5

6,690 22. 123,456×8

Ans. 86,415,047,523

15. 273×8

24. 12,345,006,789×7

25. 236,120,013×5

215,576

417,640

987,648

22,383 23. 13,052,068×5 65,260,340

1,180,600,065

Thus far, none of our multipliers have exceeded 12; and from observation, we derive the following rule.

I. PLACE THE MULTIPLIER UNDER THE MULTIPLICAND.

II. BEGINNING WITH UNITS, MULTIPLY EACH FIGURE OF THE MULTIPLICAND BY THE MULTIPLIER, AND CARRY AS IN ADDITION.

[When the pupil employs technical language, the teacher should frequently require him to explain its meaning. Thus, he should inquire the meaning of multiplier, multiplicand, &c.]

§ XII. 1. At 255 cents per pound, what cost 24 pounds of Indigo?

Here our multiplier is more than 12. Therefore, we cannot proceed by the last rule.

But we know that 24 is twice as much as 12. Therefore, 24 times any number will be twice as much as 12 times the same number.

By the last rule, 12 times 255=3,060.

But 24 times 255=twice as much, or 2 times 3,060. By the last rule again, 2 times 3,060=6,120. Therefore, 24 times 255-6,120 Ans.

2. At 16 dollars a hogshead, what cost 472 hogsheads of molasses? 16 is 4 times 4. Therefore, 16 times any number will be 4 times as much as 4 times the same number.

By the last rule, 4 times 472=1,888.

But 16 times 472=4 times as much, as 4 times 1,888. By the last rule, again, 4 times 1,888=7,552.

Therefore, 16 times 472=7,552 Ans,

Thus we multiply by a larger number, by multiplying successively by two smaller ones.

But the two smaller ones must be such that their product shall be just equal to the larger one; that is, the two smaller numbers must be factors of the larger one.

THE FINDING OF TWO OR MORE NUMBERS, WHOSE PRODUCT SHALL IS CALLED RESOLVING THE GIVEN BE EQUAL TO A GIVEN NUMBER, NUMBER INTO FACTORS.

After thoroughly understanding the foregoing, let the pupil perform the following

EXAMPLES FOR PRACTICE.

3. At 21 dollars a hogshead, what cost 284 hogsheads of molassAns. 5,964 dolls.

es?

4. What cost 28 bales of cotton at 75 dollars a bale ?

Ans. 2,100 dolls.

5. If one cask of brandy cost 15 dolls., what cost 321 ?

Ans. 4,815 dolls. Ans. 333 dolls. 18 cts.

6. If 1 pound of coffee cost 27 cents, what cost 1,234 ?

7. If 1 barrel of sugar cost 35 dolls., what cost 11,962 ?

Ans. 418,670 dolls.

8. At 42 dollars an acre, what cost 825 acres of ground?

Ans. 34,650 dolls.

9. At 81 cents per pound, what cost 1,437 pounds of tea?

Ans. 1,163 dolls. 97 cts.

10. At 63 cents per pound, what cost 127 pounds of tobacco?

11. 275×18

12. 164×45
13. 40,073X54

14. 764,131X48

15. 91,738X81 16. 3,002×28

17. 4,381X32

Ans. 80 dolls. 01 ct..

Ans. 4,950 | 18. 9,876×49 Ans. 483,924

7,380 19. 21,378×36 2,163,942 20. 209,402×72

36,678,288 21. 786,340×56

7,430,778 22. 92,810,333×48

84,056 23. 987,543,221×63 140,192 24. 9,895,461,343×49

769,608

15,076,944

So far, our examples have been performed by dividing or resolving the multiplier into two factors.

But, we may resolve it into three, four, or any number of factors, and proceed in the same way. For example, 25. How many quarts of grain in 42 sacks, cach sack containing 64 quarts?

42.

We know that 3 times 7 are 21, and that 2 times 21 are Therefore, 42 may be resolved into the factors 2, 3,

and 7...

This answer, viz. 2,688, is 42 times 64, because twice 64 multiplied by 3, (that is, taken 3 times,) is 6 times 64; and 6 times 64 multiplied by 7, (that is, taken 7 times,) is 42 times 64.

26. Find how many gallons are contained in 16 casks, each holding 58 gallons.

2 times 2 are 4; 2 times 4 are 8; and 2 times 8 are 16. We may, therefore, resolve 16 into the 4 factors, 2, 2, 2, and 2. A. 928. When the above is thoroughly understood, let the pupil proceed to the following.

EXAMPLES FOR PRACTICE.

27. At 108 dollars an acre, what cost 34,462 acres of ground? 108 3X4X9. Ans. 3,721,896. 28. At 144 dollars a pipe, what cost 615,243 pipes of wine? 144=4X4X9. Ans. 88,594,992. 29. What cost 223,332 bales of cotton at 105 dollars a bale? 105=7X3X5. Ans. 23,449,860. 30. At 64 dolls. a ton, what cost 87,543 tons of iron? 64-4X4X4. Ans. 5,602,752.

31. At 72 cents per pound, what cost 507,832 pounds of aquafortis? 72-3×3×8. Ans. 365,639 dolls. 04 cts. 32. At 56 cts. per pound, what cost 342,516 pounds of snuff®? 56=2X4X7. Ans. 191,808 dolls. 96 cts. 33. At 1 dollar and 40 cents, that is, 140 cts. per pound, what cost 41,364 pounds of tea? 140=2×2×7×5. Ans. 5,790,960 cts. 34. At 125 cents, or 1 dollar and 25 cts. per bushel, what cost 22,874 bushels of rye? 125=5X5X5. Ans. 2,859,250 cts.

35. Multiply 78,946 by 84.

36. 839,675X96.

37. 2,847,333×54.

38. 9,999,888×56,

39. 72,863,452×72.
40. 81.998,473×64.

41. 91,339,478x81.
42. 637,853,962x108.

The learner will soon find, that there are some numbers which he cannot resolve into factors. We shall teach him how to multiply by these hereafter.

A NUMBER WHICH CAN BE RESOLVED INTO FACTORS, (that is, one which is the product of two or more other numbers,) IS CALLED A

COMPOUND OR COMPOSITE NUMBER.

The factors which produce the composite number are sometimes called its component parts.

Hence, to multiply by a composite number,

RESOLVE THE MULTIPLIER INTO FACTORS, AND MULTIPLY BY THESE FACTORS SUCCESSIVELY.

[It may here be suggested, once for all, that the teacher should always require an explanation of the language employed. If the pupil hesitates, he should be referred to that part of the book which explains it.]

§ XIII. 1. What cost 13 tons of potash, at 139 dollars a ton?

Here 13, the multiplier, is more than 12, and is not a composite number.

First find, then, what 6 tons cost, thus,

139

Now 6 tons and 7 tons are 13 tons.

Therefore, add the two products together, thus, 834

973

1807

A. 5,221.

1,807 dollars, is, therefore, the price of 13 tons. 2. At 227 dollars an acre, what cost 23 acres of land? Then, when the multiplier is more than 12, and is not a composite number, we may

SEPARATE THE MULTIPLIER INTO SUCH PARTS THAT NONE OF THEM SHALL EXCEED 12, MULTIPLY BY THOSE PARTS SEPARATELY, AND ADD TOGETHER THE SEVERAL PRODUCTS.

EXAMPLES FOR PRACTICE.

3. What cost 123 barrels of flour at 17 dollars a barrel ?

Ans. 2,091 dolls.

4. At 13 cents per pound, what cost 175 pounds of sugar?

Ans. 22 dolls. 75 cts.

5. At 19 cents a quart, what cost 245 quarts of gin?

Ans. 46 dolls. 55 cts.

6. At 34 cents per pound, what cost 141 pounds of chocolate?

7. 206×39

8. 205×23

9. 876×29

10. 9,893×31 11. 7,640X37 12. 63,994×41

Ans. 47 dolls. 94 cents.

Ans. 8,034 | 13. 57,895×31

4,715 14. 893,654×23 25,404 15. 900,009×17

16. 111,222×29

17. 34,756,810X43
18. 450,000,354×47

§ XIV. The following mode, however, will be found. more convenient.

1. At 134 dollars a pipe, what cost 31 pipes of brandy? Here 31, the multiplier, is not a composite number; but 30 is, having the factors 3 and 10.

Now, if we multiply by 3 and 10 successively, we shall obtain the price of 30 pipes; to this we can add the price of 1 pipe, and we shall then have the price of 31 pipes, the answer required.

Multiply then, by 30, multiplying first by 3;
thus,

Then by 10; thus,

This is the price of 30 pipes; add the price.
of 1 more; thus,

134

3

402

10

4020

134

Therefore, 4,154 dollars is the price of 31 pipes. 4154 2. In 47 boxes, each containing 348 candles, how many candles?

Here 47, the multiplier, is not a composite number; but 45 is, having the factors 9 and 5. Ans. 16,356.

Hence, when the multiplier is not a composite number,

I. TAKE A CONVENIENT COMPOSITE NUMBER LESS THAN THE MULTI

PLIER.

II. MULTIPLY BY IT ACCORDING TO THE RULE.

III. TO THE Product, add aS MANY TIMES THE MULTIPLICAND, AS THE COMPOSITE NUMBER CONTAINS UNITS LESS THAN THE MULTIPLIER.