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of a unit of the required quantity: When the price and quantity of both commodities are given, how do you find the balance which is to be paid in money? A. Subtract what one commodity comes to, from the amount of the other. When the ready money price of one commodity, is less than its bartering price, how do you find the bartering price of the other commodity ? A. As the ready money price of the one, is to its bartering price, so is the ready money price of the other, to its bartering price.

LOSS AND GAIN, Is only a particular application of the Rule of Three Direct, by which merchants and traders discover their profits, or loss per cent, or by the gross: It also teaches them to raise or fall on the price of their goods, so as to gain or lose so much per cent, &c. Case I.--To know what is gained or lost per

.cent. RULE. First see what the gain or loss is, by subtraction; then, as the price it cost, is to the gain or loss, so is $100, or £100, to the gain or loss per cent.

EXAMPLES, 1. If a merchant buy calico, at 24 cents a yard, and sell it, at 36cts. per yard; what does he gain per cent, or what does he gain in laying out $100? Ans. $50, or 50 per cent. ats. cts. cts.

DEM.-it is evi. ,36 24:,12:: 100,00

dent, that the mer

chant on the sale ,24

12

of one yard gainGain ,12 per yd.

;24) 1200,00(50,00 ed half what it 120

cost him, conse

quently he must 000

have gained

$50,

in laying out $100; because the gain on 100 dollars must be in the same proportion as the gain on twenty-four cents.

2. Bought butter at Octs. a pound, and sold it at 12cts. a pound; what was gained per cent ? Ans. 33 per cent.

3. Bought 30 yards of broadcloth at $4,50 cents per yard, and sold it for 110 dollars ; 'what was gained or lost per çent ?

Ans. $18,51çts. 8m. loss per cent. Case II.—To know how a commodity must be sold, to gain or lose so much per cent.

RULE.-As $100, or £100, is to the price it cost, so is $100 or £100, with the profit added, or the loss subtracted, to the gaining or losing price.

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EXAMPLES. 1. If a merchant buy sugar, at 9cts. a pound, how must he sell it per pound, to gain 25 per cent ? Ans. llcts. 2m. $ cts.

DEM.--As the third term is increasÁs 100:,09 : : 125 ed by the per cent above the first, so

,09 the fourth term or answer; rust in1100), 11/25 crease above the second term. To gain

25 per cent, is adding one fourth the given sum or cost to itself.

2. If a man buy corn, at 40 cents per bushel, how must be sell it, to gain 30 per cent? Ans. 52 cents per

bushel. CASE III.—When there is gain or loss per cent, to know nchat the commodity cost.

RULE.-As $100, or £100, with the gain per cent added, or the loss per ct. subtracted, is to the price, so is 100 dollars, or 100 pounds, to the first cost.

EXAMPLES. 1. If a yard of cloth be sold at $4,50cts., and there is gain$25,50cts. per cent, what dia the yard cost?

Ans. $3,58 cents 5 mills. $25,50 100 $ cts.

125,50 4,50 :: 100,00 : Ans. $3,58cts. 5m. 2. By selling wheat, at 75 cents per bushel, I lose at the rate of 30 per cent, what was the first cost of the wheat per bushel ?

Ans. $1,07 cents. CASE IV.-If by wares sold, at a given rate, there is so much gained or lost per cent, to know what would be gained or lost per cent, if sold at another rate.

RULE.-As the first price is to $100 or £100, with the profit per cent added, or the loss per cent subtracted, so is the other price, to the gain or loss per cent, at the other price.

Note.—If the answer exceed 100, the excess is gain per cent, but if it be less than 100, the deficiency is loss per cent.

EXAMPLES. 1. If brandy, sold at $1,12 cents per gallon, be 20 per cent profit, what gain or loss per cent shall I have, if I sell the same at 95cts. per gallon ? Ans. Gain $1,78,5m. pr. ct.

$1,12 : 120.::.95 : 101,78,5

:

yard ?

per bushel ?

2. If I sell cloth at 60cts. per yard, and thereby gain 25 per cent, what shall f gain per cent, if I sell it at 7Octs. per

Ans. I shall gain $45,83cts. 3m. per cent, 3. If I sell wheat at $1,25. per bushel, and thereby gain 10 per cent, what shall I gain or lose per cent, if I sell it at 90cts.

Ans. Lose $20,80ets. per cent. $1,25 : 110 :: 90 : 79,20 QUESTIONS ON LOSS AND GAIN. What is Loss and Gain ? A. It is a particular application of the Rule of Three, by which merchants are able to discover their profit or loss per cent; it also teaches them to raise or fall on their goods so as to gain or lose so much per cent. How do you find what is gained or lost percent? A. First find what is gained or lost on a yard or pound by subtraction; then say, as the price it cost is to the given gain or loss, so is 100 dollars to the gain or loss on 100 dollars ; because it is plain, that the gain or loss on 100 dollars is in proportion to the gain or loss on the price of one yard or one pound. How do you find how a commodity must be sold to gain or lose a certain per cent ? A. As 100 dollars is to the price it cost, so is 100 dollars, with the profit added, or the loss subtracted, to the gaining or losing price; for as the third is increased by the gain, or lessened by the loss subtracted, so the fourth term or answer will become increased, or diminished, in proportion to the second term.

DISCOUNT, Is an allowance made for the payment of money before it becomes due, or upon advancing ready money on notes, obligations, &c. which are payable at some future period. What remains after deducting the discount, is called the present worth, or such a sum as, if put to interest, would, in the given time, and at the given rate per cent, amount to the sum or debt then due. This rule is only an application of the Rule of Three Direct, as may be seen from the first example.

RULE.-As the amount of 100 dollars, pr 100 pounds, for the given Fate and time, is to 100 dollars, or 100 pounds, so is the given sum or debt to the present worth. Subtract the present worth from the given sum, and the remainder will be the discount.

PROOF.–Find the amount of the present worth, for the time and rate on the given sum, which must equal the given sum or debt.

EXAMPLES 1. What is the present worth, and what is the discount of 100 dollars, payable in one year, at 7 per cent a year?

Ans.

$93,45cts. 7m. present worth

$6,54cts. 3m. discount,

107 : 100 :: 100 100

Proof: 107)10000(93,45cts. 7m. Present worth.

93,45,7
963

7
370
$100,

6,54199
321
93,45,7

93,457
490

$6,54,3 Discount. · 99,99,899 428

DEM.-It is plain that 100 dollars, the se620

cond term, is the present worth of 107 dollars, 535

due a year hence, because 100 dollars put to

interest at 7 per cent, in one year, amounts 850

to $107. And it is evident that the fourth 749 term, or answer, bears the same proportion to

100 dollars; the third term, that 100 dollars, Rem. 10.1

the second term, bears to 107 dollars, the first

term. The truth of this rule is also evident from the nature of Simple Interest; for as the debt may be considered as the amount of some principal, (called the present worth,) at a certain rate per cent, and for the given time, that amount must be in the same proportion to its principal or present worth, as the amount of any other sum, to its principal or present worth. ! Note.-In discount, money is supposed to bear no interest till after it becomes due; and that a discount should be made for the payment of such obligations, before they become dué, is very reasonable, biecause tře debtôr by retaining the money till it becomes due, may put it to interest for the time; but by paying it before it becomes due, he gives that benefit to another. Some have very erroneously supposed, that the interest on the given sum for the given rate and time, was the discount, and this interest çaken from the principal, gave the present worth ; but our example proves it to be untrue, because, according to that, the discount on 100 dollars would be 7 dollars, and the present worth of 100 dollars, due a year hence, at.7 per cent, would be $93; but 93 dollars put to interest for one year, at 7 percent, will not amount to 100 dollars, consequently they labour under a mistake who suppose the interest to be the discount.

2. What is the present worth of $500 due 2 years hence, at 7 per cent per annum?

Ans. $438,59cts. 6m. 3. What is the discount on 1000 dollars, due four years hence, at 7

per
cent?

Ans. $218,75cts. 4. What is the present worth of 1500 dollars, due 90 days hence, at 7 per cent ?

Ans. $1474,55cts, Note. When a debt is to be discharged by several payments, to be made at different times; find the present worth of each payment by its self, and the sum of hese will be the present worth of the whole.

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5. What is the preserit worth of 800 dollars, the one half payable in one year, and the other half in 'two years, at 7 per cent?

Ans. $724,70cts. 8m. QUESTIONS ON DISCOUNT. What is discount? A. It is an allowance made for the payment of money before it becoines due. On what rule is discount depending? A. On the Rule of Three Direct. What rule do you observe in sta'ting the sums in Discount? A. The sums are stated in the Rule of Three Direct, by saying, as the amount of 100 dollars, for the given rate and time, is to 100 dollars, so is the given sum to the present: worth ; and the present worth subtracted from the given sum, gives the discount. What is the present worth ? A. It is such a sum as, if put to interest, for the given time and rate, will amount to the given sąm. Is the discount on a given sum less than the interest on the given sum for the given time and rate? A. It is; because taking the interest from the given sum, for the given time, and at the given rate, the remainder put at interest for the given time and rate, would not amount to the given sum.

TARE AND TRET, Tare and Tret are allowances made to the buyer, on the weight of some particular commodities.

Tare is an allowance made for the weight of the barrel, box, bag, or whatever contains the articles or goods.

Tret is an allowance of 4lb. on every 104lb. for waste, dust, &c.

Cloff is an allowance, on some commodities, of 2lb. on every 3cwt. to turn the scale, or to inake the weight hold out, when goods are reweighed, and is claimed chiefly, or only, by the merchants of London.

Sutile is what remains after a part of the allowance is deducted from the gross weight.

Neat weight is what remains after all allowances are made.

All the questions in this rule may be worked by the Rule of Three Direct, and, like the Rule of Three, reduced back to Multiplication and Division; but it is frequently more convenient to work by taking - aliquot parts, as in Practice.

CASE I.- When the tare is given on the whole gross weight.

RULE.-Subtract the tare from the gross, and the remainder will be the neat weight.

EXAMPLES. 1. What is the neat weight of 112cwt. 3qrs. 12lb. of to. bacco, tare on the whole, 6cwt. 3qrs. 201b. ?

Ans. 105cwt. 3qrs. 201b. 112 3 12 By Compound Subtraction

6 3 20 Ans. 105 3 20 neat weight

cwt. qrs. lb.

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