SECTION LXXII. POSITION. POSITION is a method of performing such questions as cannot be resolved by the common direct rules, and is of two kinds, called single and double. Excentes, method SINGLE POSITION. serposing SINGLE POSITION teaches us to resolve those questions whose results are proportional to their suppositions. RULE. Take any number and perform the same operations with it as are desirable to be performed in the question. Then say, as the result of the operation is to the position, so is the result in the question to the number required. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, replied, that if he had as many more as he now had, and half as many more, he should have 200; of how many did his school consist? Suppose he had 60 As 150 : 200 :: 60 Then, as many more 60 Half as many more 30 150 60 150)12000 (80 scholars, Ans. 12000 By analysis. By having as many more, and half as many more, he must have 2 times the original number; therefore, by dividing 200 by 21, we obtain the answer, 80, as before. NOTE. Having performed all the following questions by position, the student should then perform them by analysis. 2. A person after spending left; what had he at first? and of his money had $60 Ans. $144. 3. What number is that, which, being increased by 1, §, and of itself, the sum shall be 125 ? Ans. 60. 4. A's age is double that of B, and B's is triple that of C, and the sum of all their ages is 140. What is each person's age? Ans. A's 84, B's 42, C's 14 years. 5. A person lent a sum of money at 6 per cent., and at the end of 10 years received the amount $560. What was the sum lent? Ans. $350. 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 $4.80 She must then have 9 yards of lining, value 2.70 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, By Analysis. As the silk and lining contain 15 yards, and cost $7.00, the average price per yard is 463; and this taken from 80 leaves 333; and 30 taken from 463 leaves 163; and as the quantity of lining will be to that of the silk as 33 to 16, 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 31 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 × 4 × 100 = 4800lbs. whole quantity on 34 acres for 4 weeks. 480031440lbs. 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. 1890-1440450lbs. = the quantity grown on an acre for 9-4-5 weeks. 4509 490lbs. acre for 1 week. 90 × 3 × 4 = 1200lbs. weeks. 4800-1200 3600lbs. acres. the quantity which grows on each quantity grown on 34 acres for 4 original quantity of grass on 3 3600÷31 1080lbs. = original quantity on 1 acre. Then, by the last condition of the question, 24 × 1080 25920lbs. = original quantity on 24 acres. 24 × 90 × 1838880lbs. quantity which grows on 24 acres in 18 weeks. 2592038880=64800lbs. = whole quantity on 24 acres for 18 weeks. 64800183600lbs. each week. quantity to be eat from 24 acres 3600 ÷ 10036 = 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. 607 seconds. (2.) Also before the minute-hand will be equally distant between the other two hands? Ans. 6188 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 |