--- 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. EXAMPLES. 1. How much tea at 9s. 6d. per lb. must be given in barter for 156 gallons of wine, at 12s. 34d. per gallon? 8. d. gals. s. d. lb. Or, As 12 31: 156: 96: 201 1354 Ans, as before. 2. A and B would barter; A has 150 bushels of wheat, at $1.25c. per bushel, for which B gives 65 bushels of barley, worth 621c. per bushel, and the balance in oats at 374c. per bushel. What quantity of oats must A receive from B? Ans. 391 bushels.. 3. A has linen cloth, at 30c. per yard, ready money, in barter ·36c.; B has 3610 yards of ribband, at .22c. 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 1980 yards of linen, and $200 in cash. 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? 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 $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 $10-20 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 per cent. a year on the purchase price? 275. What is Loss and Gain ?—276. in business?-- -277. How do you proceed, Ans. S$3.12 whole gain. 2.50 gain per cent. What is the rule for finding loss or gain when goods are bought or sold en eredit? 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 ? 2. Bought a piece of cloth at $2.75 per yard, and sold it at a loss of 15 per cent. what was it sold for per yard? 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. ? Ans. $1.03c. 14m. EQUATION OF PAYMENTS. 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 founded 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 payable after the equated time, from that to their terms; but this is not correct, for by keeping a debt unpaid after it is duc, 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; there fore, 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.? 279. What is Equation of Payments?~280. What is the rule ; and on what is it founded? EXAMPLES. 1. A owes B $380 to be paid as follows, viz. $100 in 6 months, $120 in 7 months, and $160 in 10 months: What is the equated time for the payment of the whole debt? 100X 6= 600 120X 7 840 160X10=1600 100+120+160=380)3040(8 months, Ans. 3040 2. A owes B £104 15s. to be paid in 4 months, £161 to be paid in 3 months, and £152 5s. to be paid in 5 months: equated time for the payment of the whole? What is the Ans. 4 months and 8 days. 3. There is owing to a merchant £698, 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? Ans. 4 months. 4. The sum of $164 16c. 6m. is to be paid, in 6 months, in 8 months, and in 12 months: what is the mean time for the payment of the whole ? Ans. 73 months. 5. A merchant bas $360 due him, to be paid at 6 months, but the debtor agrees to pay 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? 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. Thus 4 is the root or 1st power of 4. 4x4-16 is the 2d power, or the square of 4. =42 4x4x4 64 is the 3d power, or the cube of 4. =43 4x4x4x4=256 is the 4th power, or the biquadrate of 4,&c.=4' 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 re duce the vulgar fraction to a decimal. EVOLUTION. 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 4x4 16. 4 also is the cube root of 64, because 4x4x4-64; and 3 is the square root of 9, and 12 is the square root of 144, and the cube root of 1728, because 12x12x12=1728, and so on. 281. What is Involution ?do you find these powers? -284. -282. What are powers of numbers ?- 283. Hore What is Evolution 3283. What is a root? |