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300 | 25.
19. 20. 21. 22. 23.
894,320 90 1 24.
70,620 800,000 26.
XVII. We will now give a process, adapted to all cases in Multiplication.
1. A carpenter received 1,565 dollars for building a house. How much ought he to receive for building 17 houses at that rate ?
Here 17 is the multiplier. First find how much he ought to receive for 7 houses ; thus,
Then find how much for 10 houses, by
annexing a cypher to 1,565 ; thus,
Ans. 26605 2. In a garrison, if the allowance of one man is 23 ounces of provision a day, how many ounces a day ought 634 men to have ?
Here 23 is the multiplier.
First, find what it would amount to at 3 ounces per day; thus,
1902 Then, at 20 ounces; thus,
But, 20 and 3 are 23. Therefore,
add the two products; thus,
14582 Ans. In these examples we found the answer from multiply. ing by the units in the multiplier, (according to x1,) then, by the tens, (according to Š XVI,) and, finally, adding the two products together. We
may do the same thing a little more concisely, ihus :
3. How many yards in 29 pieces of cloth, each piece containing 132 yards.
132 Multiply first by the 9 units, and
29 then by the 2 tens.
Then, add the two products, thus, 3828 Ans.
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, omit953 ting to annex the cyphers, required by the rule
in Sect. xvi, because, in this case, unnecessary. $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
Here, the tens' order is filled by a cypher. It 1295 has, consequently, no partial product, 185
$19795 Ans. Hence, the general rule for Multiplication.
1. 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 !
Ans. $505.44. 9. A man bought 52 oxen, at $37.00 a head. What did they all come to ?
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?
Ans. $35,113. 16. 34,293X74 Ans. 2,537,682 1 18. 37,864X209 7,913,576 17. 47,042X91 4,280,822 19. 25,203X4,025 101,442,075 20. 269,181 X4,629
1,246,038,849 21. 40,634 X 42,068
1,709,391,112 22. 134,092x87,362
11,714,545,304 23. 918,273,645X1,003,245 921,253,442,978,025 In ġ x. 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 MULTIPLY AGAIN.
But this is sometimes inconvenient ; therefore, in such cases,
II. IF THE FIRST PRODUCT WAS OBTAINED BY THE RULE IN 0 X1, XV, XVI OR XVII, OBTAIN ANOTHER, BY THE RULE ÎN Ở 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 ? FIRST OPERATION.
216.25 Ans. as before.
II. At 56 dollars a barrel, what cost 129,874 barrels of brandy ?
Operation by 9 XII. Proof by & XVII.
56 posite number, whose factors 1038992
779244 are 8 and 7.
7,272,944 Ans, as before. 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 contain. ing 29 yards ?
6. A ship sailed 26 days at the rate of 125 miles a day, and after. wards, 43 days at the rate of 98 miles a day. How far did she sail in all ? 7. 72,600X31,000.
11. 38,603X950,950 8. 49,000 X 52,000.
12. 712,314x23,000,333 9. 607,080X539,063
13. 37,969,868 X 98,647,564 10. 41,725X687,900
| 14. 363,464,565X271,892,955 15. 437,920,2137234,678,956 16. 234,567,890, 123X 345,678,900,124
ġ 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.
1 shilling 20 shillings
£ 21 shillings
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 fd.; 2 farthings, bd.; and 3 farthings id.
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 1 shilling? In 2 shillings ? 5. How many pence in 1 pound ? In 2 pounds ? Ăns. 240. 480. 6. At 2 pence apiece, how many ponce will 17 oranges cost? 7. How many pence in a ĝuinea ? In 2 guineas ? Ans. 252. 504. 8. How many farthings in a pound? A.960. In l 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£. 1ls. 10d. 2qrs. ? Ans. 8,250. 15. How many qrs. in 2,853£. 198. 9d. 3 qrs. ? Ans. 2,739,831, 16. How many qrs. in 78,364£. 158. 7d. lqr. ? 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. 16 ounces
lb. 14 pounds
sto. 28 pounds
1 quarter of a hundred weight, qr, 4 qrs. or 112 lbs.
1 hundred weight, cwt. 20 hundred weight 1 ton
T. NOTE. By this weight are weighed all coarse and drossy articles, which are 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 weighi. 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. ?