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2. For value received, I promise to pay John P. Smith, or order, Eight Hundred and Seventy-five Dollars, with interest. January 10, 1821. $875.
6, 15, 1, 2
I LXXVI. PRACTICE IN COMPOUND NUMBERS.
Operations in compound numbers, as pounds, shillings, for instance, may be shortened by taking aliquot parts, as in Practice of Federal Money, 41 XXVIII.
1. What is the cost of 28 bushels of salt, at 10 s. a bushel ?
In this example, 10 s.= 4£.; then, I of 28 bushels is the cost in pounds, thus, 28-2=146., Ans.
2. What will 40 bushels of wheat cost, at 58. = 4£. a bushel ? At 10 s. - Ž£.? At 4 s. =*£.? At 1s. = =26£.?
I. Hence, when the price is an aliquot, or even, part of a pound, we divide the number of gallons, yards, &c. by this aliquot part, as in 9 XXVIII.
Exercises for the Slate,
1. What will 8640 yds. of cloth cost, at 10 s. = s£. a yd. ?4320. At 6 s. 8 d. =*£.?-2880.
At 5 5.5
=*£. ?-2160. At 48.=;£. ?-1728. At 3 s. 4 d.=*£. ?-1440. At 2 s. 6 d. = $£. ?-1080. At I s. 8 d.=17£.7-720. At 1 s. 4 d.=i}£. ?576. At 1s. 3 d.=T£, ?-540. At 18.=z£. ?-432. At 10 d.=27=24£. ?-360. At 8 d. =36£. ?-288. At 5 d.= 18£. ?-180. At 24 d.=94£.2-90. A. 16794£.
Note. The aliquot parts of a pound, in the following examples, may be found in the former examples.
2. What cost 20 gallons of brandy, at 6 s. 8 d. per gallon ?6,13,4.
3. What cost 8 yds. of broadcloth, at 10 s. per yard ?-4. 4. What cost 25 bushels of rye, at 5 s. per bushel ?-6,5. 5. What cost 30 bushels of oats, at 2 s. 6 d. per bushel ?-3,15. Ans. 20£. 13 s. 4 d.
6. What will 51 bbls. of cider cost, at 7 s. 6 d. per bbl. ?
12£. 15 s.=cost, at 5 s. per bbl.
6£. 7 s. 6 d. =cost, at 23. 6 d. per bbl. Ans. 19£. 2s. 6 d. - cost, at 78. 6 d. per bbl.
5s. = 1£.) 51
6£. 7 s. 6d.= cost, at 2 s. 6 d. per bbl. Ans. 19£. 2 s. 6 d. =cost, at 7 s. 6 d. per bhl.
II. Hence, when the price is not an aliquot part of 1£., we may first find what is the greatest even part, and then take parts of this part, and so on, for several times.
7. What will 20 yds. of cloth cost, at 12s. 6d. per yard?-12,10.
bushel ?13,10. 10. What cost 12 bbls. of ale, at 17 s. 6 d. per bbl.?-10,10.
A. 66£. 10 s. 11. What will 5 cwt. 3 qrs. 21 lbs. of sugar cost, at $9,60 per cwt. ?
? qrs.= } cwt. $9,60
12. At $2,50 per yard, what will 5 yds. 2 qrs. of broadcloth cost ?-1375. What will 4 yds. 1 qr. cost?-10625. Will 6 yds. 3 qrs. ?-16875. Ans. $41,25.
13. 5 ewt. 3 qrs. 16 lbs., at $4,20 per cwt. ?-2473.
1. Two boys, William and Thomas, trading with marbles, in company, gained 80 cents; William owned 4 of the marbles, and Thomas }; what was each one's part of the gain?
2. James and Rufus, owning a sled, in company, sold it for $3 more than it cost, that is, $3 gain; Rufus owned ; of it, and James }; what was each one's share of the gain ?
Q. What is the Rule of FELLOWSHIP?
A. When men are trading in company, it ascertains the gain or loss to be shared by each.
Q. What is called Stock, or Capitat?
1. Three men, A, B, and C, traded in company; A put in $200, B $400, and Ć $600; they gained $300: what was each man's share of the gain ?
In this example, it is evident, that B ought to have twice as much of the gain as A, for his stock is twice as much, and C 3 times as much as A ; that is, each man's gain or loss ought to have the same relation to the whole gain or loss, as the money he put in has to the money they all put in. The same principle will apply in all cases in which a number is to be divided into parts, which shall have a given relation, or ratio, to cach other, as the dividing a bankrupt's estate among his creditors, apportioning taxes, &c,
Hence, from the foregoing example, we derive the following proportions :
A's stock, $2007
1200 : 200 :: 300 : 50, A's gain. C's stock,
$600 1200 : 400 :: 300 : 100, B's gain. Whole stock, $1200
1200 : 600 :: 300 : 150, C's gain. By ratios. These are 1200, 1200, TROTS
1, 3, 1; then, 300 X 4, that is, of $300,=$50, A's; } of $300 =$100, B's, and of $300=$150, C's gain.
Or, by analysis. If $1200 stock gain $300, then $1 stock will gain 1200 of $300,= 1920 = $1 Now, if $1 gain $1, then of $200, A's stock, = $50, "A's gain ; ¢ of $400, B's stock,=$100, B's gain; and 1 *F $600, C's stock,= $150, C's
This last method will generally be found the shortest, and best adapted to business; especially when there are several statements, in which all the first terms are alike, and all the third terms are alike.
PROOF. It is plain, that, if the work be right, the amount of the shares of the gain or loss must be equal to the whole gain or loss; thus, in the last example, A's is $50+ B's $100 + C's $150= $300, the whole gain.
Q. What, then, is the RULE?
A. As the whole stock : to each man's stock :: the whole gain : to each man's gain.
2. Three merchants, A, B, and C, gained, by trading in company, $200; A's stock was $150, B's $250, and Ĉ's $400; wirat was the gain on $1, and what was each man's share of the gain? A. The gain on $1 is $!; then, & of $150= $37,50, A's; of $250 = $62,50, B's; and ļof $400 =
$100, C's. 3. A, B, and C, freight a ship with 270 tons; Å shipped o!. board 96 tons, B 72, and C 102; in a storm, the seamen were obliged to throw 90 tons overboard; what was the loss on 1 tou? and how many tons did each lose ? A. The loss on 1 ton is } of a ton; A's, 32; B's, 24; C's, 34.
4 A and B trade in company, with a joint capital of $600 ; A put in $350,50, and B $249,50, and, by trading, they gained $120: what is the gain on $1, and what is each person's share of the gain? Ans. $$; A's, $70,10; B's, $49,90.
5. A ship, valued at $25200, was lost at sea, of which belonged to A, ) to B, and the remainder to C: what is the loss on $1? and how much will each man sustain, supposing $18000 of her to be ensured ? Ans. $ ; A's, $2400; B's, $3600, and C's, $1200.
Perform the following examples, in the same manner, by finding how much it is for $1, or unity.
6. A detachment, consisting of 5 companies, was sent into a garrison, in which the duty required 228 men a day: the first company consisted of 162 inen; the second, 153; the third, 144? the fourth, 117; and the fifth, 108 ; how many men must each company furnish, in proportion to the whole number of men? A. The proportion for 1 man is }; then, of 162 =54, first company; the second, 51; the third, 48; the fourth, 39; and the fifth, 36 men.
7. Two men, A and B, traded in company, with a joint capital of $1000 : they gained 400, of which A took $300, and B the remainder : what was each person's stock? Ans. $1 gain requires 24 stock: A's, $750; B's, $250.
8. Divide $1000 between 4 persons, so that their shares may be to each other as 1, 2, 3, 4. A. $100, $200, $300, $400.
9. A bankrupt is indebted to A$350, to B $1000, to C $1200, to D $420, to É $85, to F $40, and to G $20;"his whole estate is worth no more than $1557,50 : what will be each creditor's part of the property ?
In adjusting claims of this nature, it is the general practice to find how much the debtor pays on $1, which is, in this case,
A. A, $175; B, $500; Č, $600; D, $210; E, $42,50 ; $20; G, $10. 10. A wealthy merchant, at his death, left an estate of $30000, to be divided among his 5 children, in such a manner that their shares shall be to each other as their ages, which are 7, 10, 12, 15, 16 years; what was the share of each?
Ans. $3500, $5000, $6000, $7500, $8000. 11. A and B invest equal suns in trade, and clear $220, of which A is to have 8 shares, because he spent all his time in managing the concerns, and B, only 3 shares : what is each man's gain? and how much is A allowed for his trouble? Ans. $160; A's share, $100 for his trouble; $60, B's share.
12. If a town raise a tax of $1920, and all the property in town be valued at $64000, what will that be on $1? and what will be A's tax, whose property is valued at $1200 ? sins. $,03 on a dollar; A's tax, $36.
In assessing taxes, we must first make an inventory of all the property, both real and personal, of the whole town, and also of each individual who is te be taxed; and, as the whole number of polls are rated at so much each, the tax on all the polls must first be taken out from the whole tax, and the remainder is to be assessed on the property. Then, to find how much any individual must be taxed for his property, we need only find how much the remainder of the whole tax is on $1, and multiply his inventory by it.
Note.--In some states, taxes are assessed only on the real and personal estate of the inhabitants, no poll taxes being allowed.
13. A certain town is taxed $2140; the whole property of the town is valued at $500000; "there are 200 polls, which are taxed $,70 each; A's property is valued at $1400, and he pays: for 2 polls :
I's, « 800,40,
I's, “ 375,25,
K's, " 265,30,
66 2. What will be the tax on $1? and what will be A's tax ?. 200 polls X $,70 $140, amount of the poll taxes; and $2140 – $140= $2000, which is to be assessed on the property. $500000 : $2000 :: $1 : $,04, tax on $1. Then, to find A's tax, his inventory being $1400, we proceed thus :
$1400 X $,04 =$56
$57,40, A's wliole tax, Ans.