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3. When, in bartering, one commodity is reckoned above the ready money price,—to find the bartering price of the other —Say, as the ready money price of the one, is to its bartering price ; so is that of the other, to its bartering price : Next, find the quantity required, according to either the bartering or ready money price.

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2. A and B would barter; A has 150 bushess of wheat, at $1.25c. per bushel, for which B gives 65 bushels of barley, worth 62.4c. per bushel, and the balance in oats at 37.4c. per bushel: What quantity

of oats must A receive from B 2 Ans. 391; bushels.

3. A has linen cloth, at 30c. per yard, ready money, in barter .36c.; B has 3610 yards of ribband, at 220. per yard ready money, and would have of A $200 in ready money, and the rest in linen cloth: what rate does the ribband bear in barter per yard, and how much linen must A give B?

Ans. The rate of ribband is 26c. 4m. per yard, and B must receive 19803 yards of linen, and $200 in cash. w

LOSS AND GAIN.

Loss AND GAIN is a rule by which merchants and traders find what they gain or lose by trading, and at what rate per cent. :It also teaches them to find the price for which any kind of goods must be sold, in order to gain or lose any given rate per cent.— The different cases are only particular applications of the Rule of Three.

CASE I.

When goods are bought at one price, and sold at another, to find what is gained or lost, and the gain or loss per cent.

RULE.—Find the gain or loss by subtraction ; then, As the price the goods cost, is to the gain or loss, so is $100 or £100 to the gain or loss per cent.

EXAMPLES.

1. If I buy cloth for $2 a yard, and sell it for $2.50 a yard; what do I gain per yard, and what do I gain per cent. or by laying out 100 dollars ?

$ c. $ c. $
Sold for 2.50 As 2 : 50 : : 100
Cost 2:00 * 100
Gained 50 per yard. 2)5000.

Gained $2500 per cent.

Note. When goods are bought or sold on credit, the present worth of the value of the goods for the time, must be found, in order to find the true gain or loss.

2. If I buy cloth at $3.50 per yard for cash, and sell it at $4.12 per yard on a credit of six months, what do I gain per cent. allowing discount at 6 per cent, a year on the selling price Ans. $14.284 per cent.

3. Bought 12 cwt. 2 qrs. of sugar at $1020 per cwt. on a credit of 4 months, and sold the same at $10.25 per cwt. for cash; what was the whole gain, and the gain per cent, allowing discount at 6 percent. a year on the purchase price $3.124 whole gain. 2:30 gain per cent.

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275. What is Loss and Gain 3–273. What is the rule for finding loss or gain in business 3–277. How do you proceed, when goods are bought or sold on credit *

CASE II.

To find the price for which any kind of goods must be sold, in order to gain or lose any given rate per cent.

RULE.—As $100 or £100 is to the purchase price, so is $100 or £100 with the profit per cent. added, or loss per cent. subtracted, to the selling price.

EXAMPLES.

1. Bought linen at 60 cents per yard, how must it be sold per yard, in order to gain 25 per cent.” $ C. C. As 100 : -60 : : 125 : 75 Aris.

2. Bought a piece of cloth at $2.75 per yard, and sold it at a loss of 15 percent. : what was it sold for per yard 2 Ans. $2.3375.

3. Bought 50 gallons of brandy, at 75 cents per gallon, but by accident, 10 gallons leaked out : At what rate must I sell the remainder per gallon, to gain upon the whole prime cost, at the rate of 10 per

cent. 2 Ans. $1.03c. 14m.

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EQUATION of PAYMENTs is the finding of a time to pay at once, several debts due at different times, so that neither party shall sustain loss.

RULE.—Multiply each payment by the time at which it is due ; then divide the sum of the products by the sum of the payments, and the quotient will be the equated time.*

*This rule is sounded on a supposition, that the sum of the interests of the several debts which are payable before the equated time, from their terms to that time, is equal to the sum of the interests of the debts payabie after the equated time, from that to their terms; but this is not correct, for by keeping a debt unpaid after it is due, the interest of it is gained for that time ; but by paying a debt before it is due, the payer does not lose the interest for that time, but the discount only, which is less than the interest; therefore, the rule is not accurately true; however, in most questions which occur in business, the errour is so trifling, that it will generally be made use of as the most eligible method.

278. How do you ascertain at what price you must sell an article in order to gain so much per cent. 2–279 What is Equation of Payments 3–280. What is the rule: and on what is it founded ?

EXAMPLEs.

1. A owes B $330 to he paid as follows, viz $100 in 6 months, $123 in 7 months. and $100 in 19 months: What is the equated tale for the payment of the whole debt? 1942 to 600 12 x 7= 8.10 100X 10–1600

100+120+160-320)394 (3 months, Ans.
3949

2. A owes B £104 15s. to be paid in 43 months. £151 to be paid in 34 months, and £1525s, to be paid in 5 months: What is the equated time for the payment of the whole 2 Ans. 4 months and 8 days.

3. There is owing to a merchant £998. to be paid £178 ready money. £200 at 3 months. and £320 in 8 months; I demand the indifferent time for the payment of the whole 2 Ans. 43 months.

4. The sum of $164 16c. 6m. is to be paid, 3 in 6 months, 4 in 8 months, and 4 in 12 months: what is the mean time for the payment

of the whole 2 Ans. 73 months.

5. A merchant has $360 due him, to be paid at 6 months, but the . debtor agrees to pay 3 at the present time, and # at 4 months; I demand the time he must have to pay the remainder, at simple interest, so that neither party may have the advantage of the other ?

3=180 paid down.

#=120 paid at 4 months.

#= 60 unpaid. Now as he pays 180 dollars 6 months, and 120 dollars 2 months before they are respectively due, say, as the interest of 60 dollars for 1 month, is to 1 month, so is the sum of the interest of 180 dollars for 6 months, and of 120 dollars for 2 months, to a fourth number, which added to the 6 months, will give the time for which the 60 dollars

ought to be retained.
Ans. 28 months.

INVOLUTION.

INvolution is the method of finding the powers of numbers.

Powers of numbers are the products arising from the continual multiplication of numbers into themselves.

Any number may itself be called the root or first power. If the first power be multiplied by itself, the product is called the second power, or the square ; if the square be multiplied by the first power, the product is called the third power, or the cube ; if the cube be multiplied by the first power, the product is called the fourth power, or the biquadrate, &c.

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The small figure points out the order of the power, and is called the Index or Exponent. .

Rule for finding the powers of numbers.

Multiply the given number, or first power continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required.

Note—The powers of vulgar fractions are found by raising each of their terms to the power required. If the power of a mixed number be required, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal.

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Evolution, or the extraction of roots, is the operation by which we find any root of any given number.

The root is a number whose continual multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or 2d, 3d, 4th root, &c. accordingly as it is, when raised to the 2d, 3d, 4th, &c., power, equal to that power. Thus, 4 is the square root of 16, because 4×4=16. 4 also is the cube root of 64, because 4×4×4=64; and 3 is the square root of 9, and 12 is the square root of 144, and the cube root of 1728, because 12×12×12=1728, and so on.

281. What is Involution ?—232. What are powers of numbers ?–283. How do you find these powers ?–384. What is Evolution ?—285. What is a root 3

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