LOSS AND GAIN. Loss and Gain is a rule by which persons in trade are able to discover their profit or loss; and to increase or lessen the prices of their goods so as to gain or lose on them to any given amount. Questions in Loss and Gain are solved by the Rule of Three, or by Practice. EXAMPLES. 1. A merchant bought 100 yards of silk at 75 cents per yard, what will be his gain in the sale, if he sell it for 90 cents yer yard? 15 gain per yard. As 1 yard. yards. cts. dolls. 15 15 Ans. 2. If a grocer buy 250 lbs. of tea, at $225, and sell the whole at $1.25 per lb. what will be his gain by the transaction? Ans. $87.50. 3. If a yard of calico cost 28 cents, and is sold for 31 cents, what is the gain on 293 yards? Ans. $8.79. 4. Bought 420 bushels of corn at 25 cents per bushel, and sold the same at 38 cents per bushel; what was the amount gained? Ans. $54.60. 5. A merchant bought 12 cwt. of coffee at 26 cents per lb. and afterwards was obliged to sell it at 20 cents per lb. what was his loss? Ans. $80.64. 6. If a merchant gain $80 on $560, what is that per cent.? Ans. 14 per cent. 7. If a yard of velvet be bought for 16s. and sold again for 12s. what is the loss per cent.? Ans. 25 per cent. 8. A merchant bought 2 hhds. of wine, containing 126 gals., at $1.75 a gal. and retailed the same at $2.12 a gal.: what did he gain in the whole? Ans. $47.25. 9. A merchant bought 2 pieces of broad-cloth, containing 56 yds., at $4.75 a yard; but upon examination, found them damaged. He was, therefore, obliged to sell them for $4.124 a yard; how much did he lose by the bargain? Ans. $35. 10. A gentleman purchased 1500 lbs. of coffee for $172.50, how must he sell the same to gain $32 by his bargain? Ans. 13 cts. 6 m. 33. 11. A merchant bought 250 bbls. of flour at $4.50 a bbl.; how must he sell the same to gain $55 by the bargain? Aus. $1.72. 12. A lady purchased a quantity of millinery, for which gave $184; and sold the same for $210; how much did she gain per cent.? Ans. 14.13+per cent. she INVOLUTION, OR THE RAISING OF POWERS. The product of any number multiplied by itself any given number of times, is called its power, as in the following example. Thus, 2x2-4 the square, or second power of 2. 2×2×2=8 the cube, or third power of 2. 2×2×2×2=16 the biquadrate, or fourth power of 2.* Hence, 3 rised to the 4th power makes 81. The number which denotes a power is called the index, or exponent of that power. When a power of a vulgar fraction is required, it is only necessary to raise, first the numerator, and then the denominator to the given power, and place the product of the one over the product of the other, thus, the 3d 2×2×2= power of 3X3X3= *Any given number is co sidered the first power of itself, and when multiplied by itself the product is the second power, &c: EVOLUTION, OR THE EXTRACTION OF ROOTS. The root of a number, or power, is any number, which being multiplied by itself a certain number of times, will produce that power; and is called the square, cube, biquadrate root, &c. according to the power to which it belongs. Thus, 3 is the square root of 9, because when multiplied by itself, it produces 9; and 4 is the cube root of 64, because 4x4x4-64, and so of any other number. THE SQUARE ROOT. Extracting the square root of a number, is the taking a smaller number from a larger, and such as will, being multiplied by itself, produce the larger number. RULE. 1. Separate the sum into periods of two figures each beginning at the right hand figure. 2. Seek the greatest square number in the left hand period; place the square, thus found, under that period, and the root of it in the quotient. Sabtract the square number from the first period; to the remainder bring down the next period, and call that the resoivend. 3. Double the quotient, and place it on the left hand of the resolvend for a divisor. Seek how often the divisor is contained in the resolvend, omitting the units figure, and set the answer in the quotient, and also on the right hand side of the divisor. Then multiply the divisor, including the last added figure, by that figure, that is, by the figure last placed in the quotient; place the product under the resolvend, subtract it, and to the remainder bring down the next period, if there be any more, and proceed as already directed. If there be a remainder after the periods are all brought down, annex cyphers, two at a time, for decimals, and proceed till the root is obtained with sufficient exactness. Note. When a sum in the Square Root consists of whole numbers and decimals, point off the whole numbers as above directed, then point the decimal part, 3 commencing at the decimal point and forming periods of two figures each towards the right, observing when there is only one figure left for the last period, to add a cvpher to the right of it, to make an even period.— When the sun consists entirely of decimals, separate the periods after the same manner. If it le required to extract the square root of a vulgar fraction, reduce it to its lowest terms; then extract the root of the numerator for the numerator of the answer, and the root of the denominator for the denominator of the answer. If the fraction be a surd, that is, a number whose root can never he exactly found, reduce it to a decimal, and then extract the root from it; and if the sum be a mixed number, the root may be obtained in the same way. PROOF. Square the root, adding the remainder, (if any,) and the result will equal the given number. EXAMPLES. 1. What is the square root of 20857489? Root. 20857489(1567 Answer. 16 divisor 85) 185 resolvend. divisor 906)5074 resolvend. divisor 9127)53889 resolvend, 2. What is the square root of 294849? ? Ans. 543. Ans. 6422. Ans. 4.16. Ans. .027. Ans. 2.23606. Ans. Ans. 4.168333. of 7056 men; how many 9. A general has an army must he place on a side to form them into a compact square? Ans. 84. 10. If the area of a circle be 184.125, what is the side of a square that shall contain the same area? Thus, 184.125=13.569+Answer. 11. If a square piece of land contain 61 acres and 41 square poles, what is the length of one of its sides? A. P. Thus, 61 41-9801 square poles. Then, 9801-99 rods, or poles in length, Answer 12. There is a circle whose diameter is 4 inches; what is the diameter of a circle 3 times as large? Thus, 4X4-16; and 16x3=48 and /48=6.928 +inches. Ans. 13. There is a circle whose diameter is 8 inches; what is the diameter of a circle which is only one fourth as large. 8X8-64; and 64÷4-16; and /16-4 inches. Ans. 4 inches. The square of the longest side of a right angledtriangle, is equal to the sum of the squares of the other two sides; therefore, the difference of the squares of the longest side, and either of the other sides, is the square of the remaining Iside. 14. The wall of a certain city is 20 feet in height, it is surrounded by a ditch 20 feet in breadth; what must be the length of a ladder, to reach from the outside of the ditch to the top of the wall? Ans. 281 feet. 15. On the margin of a river 24 yards wide, stands a tree; from the top of which a line 36 yards long, will reach to the other side of the stream; what is the height of the tree? Ans. 26.83+yards. 16. Two ships sail from the same port; one sails due east 50 miles, and the other due south 84 miles; how far are they from each other? Ans. 97.75 miles. 17. A ladder or pole, 40 feet long, placed in the middle of a street, will reach a window of a house on each side of the street 24 feet from the pavement; what is the width of the street? Ans. 64 feet wide. |