6. Seven eighths of a certain number exceed by 81; what is the number? Ans. 120. 7. What number is that whose exceed by 217? DOUBLE POSITION.* Ans. 87. DOUBLE POSITION teaches to resolve questions, by making two suppositions of false numbers. Those questions in which the results are not proportional to their positions belong to this rule. RULE. Take any two convenient numbers, and proceed with each according to the conditions of the question. Find how much the results are different from the result in the question. Multiply each of the errors by the contra supposition, and find the sum and difference of the products. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. If the errors are unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer. NOTE. The errors are said to be alike when they are both too great, or both too small; and unlike when one is too great and the other too little EXAMPLES. 1. A lady purchased a piece of silk for a gown at 80 cents per yard, and lining for it at 30 cents per yard; the gown and lining contained 15 yards, and the price of the whole was $7.00. How many yards were there of each? Suppose 6 yards of silk, value She must then have 9 yards of lining, value Sum of their values, $4.80 2.70 $7.50 Which should have been 7.00 So the first error is 50 too much, +.50 Again; suppose she had 4 yards of silk, value $3.20 Then she must have 11 yards of lining, value 3.30 *This rule is founded on the supposition, that the first error is to the second as the difference between the true and first supposed number is to the difference between the true and second supposed number. When this is not the case, the exact answer to the questions cannot be found by this rule. First supposition multiplied by last error, Last supposition multiplied by first error, Add the products, because unlike, 500-50 +50 = 5 yards of silk, Ans. 15-510 yards of lining, By Analysis. As the silk and lining contain 15 yards, and cost $7.00, the average price per yard is 462; and this taken from 80 leaves 331; and 30 taken from 463 leaves 163; and as the quantity of lining will be to that of the silk as 33 to 163, it is therefore evident that the quantity of lining is twice the quantity of silk. Wherefore, if 15, the number of yards, be divided into three parts, two of those parts (10) will be the number of yards for the lining, and the other part (5) will be the yards for the silk, as before. NOTE. The student should perform each question by analysis. 2. A and B invested equal sums in trade; A gained a sum equal to of his stock, and B lost $225; then A's money was double that of B's. What did each invest? Ans. $600. 3. A person being asked the age of each of his sons, replied, that his eldest son was 4 years older than the second, his second 4 years older than the third, his third 4 years older than the fourth, or youngest, and his youngest half the age of the oldest. What was the age of each of his sons? Ans. 12, 16, 20, and 24 years. 4. A gentleman has two horses and a saddle worth $50. Now, if the saddle be put on the first horse, it will make his value double that of the second horse; but if it be put on the second, it will make his value triple that of the first. What was the value of each horse? Ans. The first $30, second $40. 5. A gentleman was asked the time of day, and replied, that of the time past from noon was equal to of the time to midnight. What was the time? Ans. 12 minutes past 3. 6. A and B have the same income. A saves of his, but B, by spending $100 per annum more than A, at the end of 10 years finds himself $ 600 in debt. What was their income? Ans. $480. 7. A gentleman hired a laborer for 90 days on these conditions that for every day he wrought he should receive 60 cents, and for every day he was absent he should forfeit 80 cents. At the expiration of the term he received $33. How many days did he work, and how many days was he idle? Ans. He labored 75 days, and was idle 15 days. The following question, with some variation in the language, is taken from Fenn's Algebra, page 62. It is believed, however, that Sir Isaac Newton was the author of it. 8. If 12 oxen eat 34 acres of grass in 4 weeks, and 21 oxen eat 10 acres in 9 weeks, how many oxen would it require to eat 24 acres in 18 weeks, the grass to be growing uniformly? Ans. 36 oxen. OPERATION BY ANALYSIS. Each ox eats a certain quantity in each week, which we may suppose to be 100 pounds; and of the whole quantity eaten in each case, a part must have already grown during the time of eating. Then, by the first conditions of the question, 12 X 4 X 100 = 4800lbs. 4 weeks. 4800÷31=1440lbs. whole quantity on 3 acres for whole quantity on 1 acre for 4 weeks. By the second conditions of the question, 21 × 9 × 100 = 18900lbs. = whole quantity on 10 acres for 9 weeks. 18900÷10=1890lbs. — whole quantity on 1 acre for 9 weeks. = 1890-1440 450lbs. the quantity grown on an acre for 9-4-5 weeks. 450÷9-490lbs. the quantity which grows on each acre for 1 week. 90 × 3 × 4= 1200lbs. = quantity grown on 34 acres for 4 weeks. 4800 1200=3600lbs. original quantity of grass on 3 acres. -- 360031 1080lbs. = original quantity on 1 acre. 24 × 1080=25920lbs. = original quantity on 24 acres. acres in 18 weeks. 2592038880=64800lbs. for 18 weeks. 64800 18 3600lbs. each week. = quantity which grows on 24 whole quantity on 24 acres quantity to be eat from 24 acres 3600 100 36 = number of oxen required to eat the whole, and the answer to the question. 9. There is a fish whose head weighs 15 pounds, his tail weighs as much as his head and as much as his body, and his body weighs as much as his head and tail. What was the weight of the fish? Ans. 72lbs. 10. Suppose a clock to have an hour-hand, a minute-hand, and a second-hand, all turning on the same centre. At 12 o'clock all the hands are together and point at 12. (1.) How long will it be before the second-hand will be between the other two hands, and at equal distances from each ? Ans. 6078 seconds: (2.) Also before the minute-hand will be equally distant between the other two hands? Ans. 6168 seconds. (3.) Also before the hour-hand will be equally distant between the other two hands? Ans. 59 seconds. SECTION LXXIII. EXCHANGE. EXCHANGE is the act of paying or receiving the money of one country for its equivalent in the money of another country, by means of Bills of Exchange. This operation, therefore, comprehends both the reduction of moneys and the negotiation of bills. It determines the comparative value of the currencies of all nations, and shows how foreign debts are discharged, loans and subsidies paid, and other remittances made from one country to another, without the risk, trouble, or expense of transporting specie or bullion. BILLS OF EXCHANGE. A Bill of Exchange is a written order for the payment of a certain sum of money, at an appointed time. It is a mercantile contract, in which four persons are mostly concerned; viz. 1. The drawer, who receives the value, and is also called the maker and seller of the bill. 2. The person upon whom the bill is drawn is called the drawee. He is also called the acceptor, when he accepts the bill, which is an engagement to pay it when due. 3. The person who gives value for the bill, who is called the buyer, taker, and remitter. 4. The person to whom it is ordered to be paid, who is called the payee, and who may, by indorsement, pass it to any other person. Most mercantile payments are made in Bills of Exchange, which generally pass from hand to hand, until due, like any other circulating medium; and the person who at any time has a bill in his possession is called the holder. When the holder of a bill disposes of it, he writes his name on the back, which is called indorsing; and the payee should be the first indorser. If the bill be indorsed in favor of any particular person, it is called a special indorsement; and the person to whom it is thus made payable is called the indorsee, who must also indorse the bill if he negotiates it. Any person may indorse a bill, and every indorser (as well as the acceptor, or payee) is a security for the bill, and may therefore be sued for payment. The term of a bill varies according to the agreement between the parties, or the custom of countries. Some bills are drawn at sight; others, at a certain number of days, or months, after sight or after date; and some, at usance, which is the customary or usual term between different places. Days of grace are a certain number of days granted to the acceptor, after the term of a bill is expired. Three days are usually allowed. In reckoning when a bill, payable after date, becomes due, the day on which it is dated is not included; and if it be a bill payable after sight, the day of presentment is not included. When the term is expressed in months, calendar months are understood; and when a month is longer than the preceding, it is a rule not to go in the computation into a third month. Thus, if a bill be dated the 28th, 29th, 30th, or 31st of January, and payable one month after date, the term equally expires on the last day of February, to which the days of grace must, of course, be added; and therefore the bill becomes due on the 3d of March. Form of a Bill of Exchange. Exchange for £5,000 sterling. Boston, September 25, 1835. At ninety days' sight of this, my first Bill of Exchange (second and third of the same tenor and date unpaid), pay to James Ayer, or order, five thousand pounds sterling, with or without further advice. Messrs. Dana & Hyde, Merchants, Liverpool. John L. French. |