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3. How many yards in 29 pieces of cloth, each piece containing 132 yards,

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3828 Ans.

Then, add the two products, thus,

It will be seen that the cypher need not be placed at the right of the tens' product, if the first figure of that product be written under the order of tens. For it counts nothing in the addition.

We may proceed in the same manner, if the multiplier consists of more figures. For example:

4. What are 953 sheep worth, at $1.75 apiece? Operation. 953

175

Here, we multiplied first by the 5 units, then, by the 7 tens, and then by the 1 hundred. We 4765 likewise placed the right hand figures of the last 6671 two products, under their respective orders, omitting to annex the cyphers, required by the rule in Sect. XVI, because, in this case, unnecessary.

953

$1667.75 Thus we have a product corresponding to each order of the multiplier. These products are sometimes called PARTIAL PRODUCTS; their amount when added, the TOTAL product.

If any order in the multiplier is filled by a cypher, there will be, of course, no partial product, corresponding to that order. Thus, 5. A man gave 185 dollars a pipe, for 107 pipes of wine. What cost the whole ?

Operation. 185

107

1295

185

Here, the tens' order is filled by a cypher. It has, consequently, no partial product.

$19795 Ans.

Hence, the general rule for Multiplication.

I. PLACE THE MULTIPLIER UNDER THE MULTIPLICAND SO THAT THE SAME ORDERS MAY STAND UNDER EACH OTHER.

II. MULTIPLY THE MULTIPLICAND BY EACH SIGNIFICANT FIGURE IN THE MULTIPLIER, PLACING THE FIRST FIGURE OF EACH PARTIAL PRODUCT UNDER THE ORDER OF ITS MULTIPLIER. III. ADD THE SEVERAL PRODUCTS AS THEY STAND. THE AMOUNT WILL BE THE PRODUCT REQUIRED.

NOTE. You will be sure to arrange your partial products correctly, if you place the right hand figure of each, exactly under the figure by which you are multiplying.

EXAMPLES FOR PRACTICE.

6. What will 15 tons of hay come to, at $20.78 pr. ton?

Ans. $311.70

7. What will 857 barrels of pork come to, at $18.93 pr. barrel?

Ans. $16,223.01

8. What cost 144 bushels of wheat, at $3.51 pr. bushel?

9. A man bought 52 oxen, at $37.00 a head.

come to ?

Ans. $505.44. What did they all Ans. $1,924.00.

10. At 15 cents pr. pound, what cost 600 pounds of sugar?

Ans. $90.00.

11. What cost 33 bushels of rye, at 94 cts. pr. bushel?

Ans. $31.02.

12. If a gallon of rum cost 58 cts., what cost 203 gallons?

Ans. $117.74.

13. If 1 pound of snuff cost 47 cts., what cost 314?

Ans. $147.58.

14. If a bushel of rye cost 75 cts., how much will 829 bushels cost?

Ans. $621.75.

15. If an acre of land cost 37 dollars, what will 949 acres cost?

16. 34,293X74 Ans. 2,537,682 | 18. 37,864×209 17. 47,042 91

Ans. $35,113. 7,913,576

4,280,822 19. 25,203×4,025 101,442,075

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In § XI. it is shown that the product is the same, whichever of the factors is made the multiplier. Hence, in order to prove multiplication,

I. INVERT THE ARRANGEMENT OF THE FACTORS, AND MULTI

PLY AGAIN.

But this is sometimes inconvenient; therefore, in such cases,

II. IF THE FIRST PRODUCT WAS OBTAINED BY THE RULE IN XI, XV, XVI OR XVII, OBTAIN ANOTHER, BY THE RULE IN & XII, XIII OR XIV; AND IF THE FIRST WAS OBTAINED BY ONE OF THE LATTER RULES, OBTAIN ANOTHER, BY ONE OF THE FORMER. It is hardly necessary to illustrate these modes of proof. We will, however, give one example of each.

I. At $1.25 a pound, what cost 173 pounds of tea?

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II. At 56 dollars a barrel, what cost 129,874 barrels of brandy?

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EXAMPLES FOR PRACTICE.

1. At 27 dollars a piece, what cost 234 pieces of broadcloth ? 2. At 52 dollars a barrel, what cost 72 barrels of brandy?

3. What will 763 hundred weight of sugar cost, at 15 dollars a hundred weight?

4. At 35 dollars a ton, what cost 749 tons of iron?

5. At 33 cts. pr. yard, what cost 16 pieces of cloth, each containing 29 yards?

6. A ship sailed 26 days at the rate of 125 miles a day, and afterwards, 43 days at the rate of 98 miles a day. How far did she sail in all ?

7. 72,60031,000.

8. 49,000X52,000. 9. 607,080539,063 10. 41,725×687,900

11. 38,603×950,950

12. 712,314×23,000,333

13. 37,969,868×98,647,564 14. 363,464,565 271,892,955 15. 437,920,213×234,678,956 16. 234,567,890,123×345,678,900,124

§ XVIII. We have seen in a preceding section, that money is reckoned in the United States in Dollars, dimes, cents, &c. These are the denominations of Federal Money. In other countries, it is reckoned differently. It is important that we should understand the English mode of reckoning money, because there are various currencies of money, as they are called, in this country, whose denominations are the same, though different in value. The following are the denominations of ENGLISH,

or,

STERLING MONEY.

4 farthings (marked qr.) make 1 penny, (marked) d.

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1 shilling
1 pound

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66

1 guinea.

NOTE. £ is the mark for pounds. When it is placed before the pounds, the marks for the lower denominations are omitted, but when after, they must be employed. Thus, £3; 7; 9; 2 is written, instead of 3.£. 7s. 9d. 2qrs. 1 farthing is sometimes written d.; 2 farthings, d.; and 3 farthings d.

EXAMPLES FOR PRACTICE.

1. How many farthings in 6 shillings? In 5 shillings? In 7? 2. How many shillings in 2 pounds? In 3 pounds? In 4 pounds? 3. How many pence in 2 shillings? In 3 shillings? In 4? In 5 ? 4. How many farthings in I shilling? In 2 shillings?

5. How many pence in 1 pound? In 2 pounds? Ans. 240. 480. 6. At 2 pence apiece, how many pence will 17 oranges cost? 7. How many pence in a guinea? In 2 guineas? Ans. 252. 504. 8. How many farthings in a pound? A. 960. In 1 guinea? A. 1,008. 9. How many shillings in 275 pounds? Ans. 5,500.

10. How many farthings in 341 guineas? Ans. 343,728. 11. How many pence in 3 pounds, 5 shillings, and 6 pence? NOTE. First find the pence in 3 pounds, then in 5 shillings, then add these to 6 pence. Ans. 786.

12. How many shillings in 27 guineas, and 18 shillings?

13. How many farthings in 4 pounds, 9 shillings, and 5 pence? 14. How many farthings in 8. 11s. 10d. 2qrs.? Ans. 8,250.

15. How many qrs. in 2,853£. 19s. 9d. 3 qrs.? Ans. 2,739,831. 16. How many qrs. in 78,364£. 15s. 7d. 1qr.?

We will now give some tables of Weights and Measures. The following are the denominations of

AVOIRDUPOIS WEIGHT.

16 drams (dr.) make 1 ounce, marked oz.

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1 pound

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28 pounds

66

4 qrs. or 112 lbs.

20 hundred weight

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NOTE. By this weight are weighed all coarse and drossy articles, which aro liable to waste, as groceries, flour, hay, tallow, &c.; and likewise all metals, except those usually called the precious metals; viz. gold and silver. It will be observed that 112 lbs. make I cwt. But merchants in our large towns, particularly in seaports, allow only 100 lbs. to the cwt.

Merchants are accustomed to make allowances upon goods, bought and sold by Avoirdupois Weight, for the weight of the box, cask, or bag, which contains them; and also for waste, dust, &c. The weight, before these allowances are deducted, is called gross weight; after they are deducted, it is called net weight. In some Arithmetics, rules are given for deducting these allowances, under the head of TARE AND TRETT, which are the names, given to certain deductions. These rules, however, are superfluous, for the student, familiar with principles, will easily make them for himself.

17. How many qrs. in 5 cwt.? How many in 8? In 6? In 12? 18. How many oz. in 2 lb.? In 3? In 4? In 5?

19. How many lbs. in 5 cwt.?

Ans. 560.

20. How many drs. in 3 lbs. 2 oz.?

Ans. 800.

NOTE. First find the drs. in 3 lbs., then in 2 oz. and add them together.

21. How many lbs. in 7 T. 19 cwt. ?

Ans. 17,808. 22. How many oz. in 13 T. 15 cwt. 3 qrs. 26 lbs.? Ans. 494,560. 23. How many drs. in 25 T. 17 cwt. 2 qrs. 13 lbs. 14 oz.? Ans. 14,841,312. 24. How many qrs. in 32 T. 11 cwt. 1 qr. 2 lbs. 1 oz. 2 drs. ? 25. How many qrs. in 1,827 T. 19 cwt. 3 qrs. 15 lbs. 15 oz. 15 drs.? The following are the denominations of

TROY WEIGHT.

24 grains (gr.) make 1 penny-weight, dwt.
20 penny-weights 1 ounce

12 ounces

1 pound

Oz.

lb.

NOTE. By this weight are weighed jewels, the precious metals, coin, bread and liquors. The quality, fineness, or purity of gold, is estimated in carats, by which term is meant the 24th part of any thing. Pure gold is said to be 24 carats fine; but pure gold has been found to be too soft for the purposes of coinage, since it wears away too rapidly. The law, therefore, allows and requires that there should be mingled in all gold coined, two carats, that is two 24th parts, of alloy; by which is meant some baser metal, usually copper. This is standard gold, and is, of course, 22 carats fine. Standard silver for coinage, is not estimated by carats. Pure silver is said to be 12 oz. fine; 18 dwts. of alloy are directed to be mixed with 11 oz. 2 dwts. of pure silver, so that the compound weighs a pound.

If any

The fineness of silver and gold is tried in the fire. If they lose nothing in the trial, they are pure; the gold is 24 carats, and the silver 12 oz. fine. thing is lost, it indicates that there was alloy in the metal, and the loss must be deducted, in order to ascertain the fineness.

26. How many grains, in 2 dwts. ?
27. How many dwts. in 3 oz. ?
28. How many dwts. in 1 lb. ?
29. How many grs. in 1 lb.?

Ans. 5,760.

Ans. 48.

Ans. 60. Ans. 240.

In 2 lbs.? Ans. 11,520.

Ans. 1,695.

Ans. 479.

30. How many dwts. in 7 lb. 15 dwts. ? 31. How many grs. in 19 dwts. 23 grs.? 32. How many grs. in 37 lbs. 12 dwts. 13 grs.? Ans. 213,421. 33. How many grs. in 273 lbs. 5 oz. 3 dwts. 6 grs.?

34. How many grs. in 1,236 lbs. 11 oz. 19 dwts. 23 grs. ?

35. How many oz. in 2,325,856 lbs.? How many dwts.? How many grs.?

?

36. How many grs. in 75,863,294 lbs, 3 oz. 5 dwts. 17 grs. Ans. 436,972,575,017.

The following are the denominations of

APOTHECARIES' WEIGHT.

20 grains (gr.) make 1 scruple, marked sc. or

3 scruples

66

1 dram

66 dr. or 3

8 drams

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