DOUBLE POSITION Teaches to solve questions, by making two supposed numbers. Rule. Take any two convenient numbers, and proceed with each, according to the conditions of the question; then find how much the results are different, than the results in the question. Multiply the first supposed number by the last error, and the last supposed number by the first error. If the errors are alike, divide the difference of their products by the difference of the errors, and the quotient will be the answer. If the errors are unlike, divide the sum of their products by the sum of the errors, and the quotient will be the answer. The errors are alike, when they are both too great, or both too small; and unlike, when one is too great, and the other too small, EXAMPLES. 1. A, B and C, would divide 400 dollars between them, $0 as B may have 12 dollars more than A, and C 16 dollars more than B. I demand how much each man must have ? 2. A man, lying at the point of death, said he had in a certain chest 400 dollars, which he bequeathed to his three sons, after this manner: the first must have a certain sum, the second twice as much, wanting 32 dollars, and the third must have three times as much as the first, wanting 60 dollars. I demand how much each man gets ? Ans. The first 882, second $132, third $186. 3. A, B and C, built a house, which cost 400 dollars, of which A paid a certain sum, B paid 40 dollars more than A, and C paid as much as A and B. I demand each man's share of the building ? Ans. A paid 80 dolls. B 120, and C 200. 4. A laboring man hired with a farmer for 60 days, upon this condition, that for every day he worked, he should receive 80 cents, and for every day he was idle, he should pay 40 cents for his boarding. At the expiration of the time, he received 30 dolls. How many days did he work, and how many was he idle ? Ans. Wrought 45 days, and idle 15. 5. Divide 15 into two such parts, so that when the greater is multiplied by 4, and the lesser by 16, the products will be equal. Ans. 12 and 3. 6. Two persons, A and B, have both the same income. A saves of his yearly; but B, by spending 150 dollars per annum more than A, at the end of 8 years finds himself 400 dollars in debt. What is their income, and what does each spend per year? S Their income is 500 dollars a year. A spends 400 dollars, and B 550. { 80 A spends 360 510 x 8 = 4080 480 80 error. Ans. S500 = 55 100, which he saves. 550 x 8 = 4400 dollars, spent in 8 years. In debt $ 400 at the expiration of 8 years. 7. A certain man having driven his swine to market, viz. hogs, sows and pigs, received for them all 200 dollars, being paid for every hog 3 dollars 60 cents, for every sow 3 dolls. 20 cts. and for every pig 40 cents. There were as many hogs as sows, and for each sow there were three pigs. I demand how many there were of each sort? Ans. 25 hogs, 25 sows, and 75 pigs. 8. What number is that, which being increased by its , its 4, and 18 more, will be doubled ? Ans. 72. 9. A person being asked the hour of the day, answered, that the time past, since noon, was equal to of the time remaining until midnight. Required the time? Ans. 36 minutes past 1. 10. A gentleman has two horses, and a cart worth forty dollars, which cart, if put to the first horse, will make his value double to that of the second; but if the cart be put to the second horse, it will make his value triple to that of the first horse, What is the value of each horse ? Ans. First horse $ 24, second $32. Sappose the first horse worth $18 Value second horse 5 29 The cart 40 Cart 40 X DC 15 10 800 - 180 = 120; 5 = 24 the 300 180 price of the first horse. Then $ 24 the price of the first horse: 40 the of the cart. 64;2=32 dollars, the price of the second horse, PROOF. ist horse 24 + 40 = 64 ; 2= 32 = the 2nd horse, = the 1st horse. = 24 11. There is a fish, whose head is nine inches long, and bis tail is as long as his head, and half his body, and his body is as long as the head and tail. What is the whole length of the fish? Ans. 72 inches. 12. A surly old fellow being demanded the ages of his four children answered, you may go and look; but if you must know, my first son was born just one year after I was married to his mother, who, after his birth, lived five years, and then died in childbed, with my second son. Four years after that, I married again, and within two years, had my third and fourth sons at a birth, the sum of whose two ages is now equal to that of the eldest. I demand their several ages? Ans. The first son was 22 years old, the second 17, the third il, and the fourth 11, A useful Rule. Any question, however intricate, and though incapable of being answered by common Position, may be solved by the following rule: "Proceed as in Double Position, only let your supposed numbers be as near the truth as possible, and then, by the rule of Proportion, say, as the difference of the two errors, if alike, or their sum, if unlike, is to the ifference of the two positions, so is the least error to the difference between its position, and the first approximate value of the number sought. Repeat the operation with this approximate value, and the position which gave the least error, for a further approximation, and so on, to any degree of accuracy required. EXAMPLE. 13. Let it be required to divide 100 into two such parts, that the product of the whole, into one of the parts, may be equal to the square of the other part ? 1st. Suppose 38 and 62. Error in this, 44- 67} 38.2--38.2 Errors 44+.76544.76; then say, as 44.76 : 2 :: 76 to .034, which is the difference between its position, (38.2,) and the first approximate value of the number sought. Therefore it will be 38.2 -.034 = 38.166, the approximate value of the least part, and very near the truth. By repeating the operation, you will come to the exact number, which, taken from 100, will leave the greatest number. 14. Two merchants put equal sums in trade: A gained a sum equal to one-fourth of his stock, and B lost 225 dolls. Then A's gain and stock was double that of B's. What suin did each put in? Ans. 600 dollars. 15. A gentleman had two silver cups, of unequal weight, having one cover to both, 5 oz. Now if the cojer is put on the less cup, it will make it double the weight of the greater cup; but if put on the greater cup, it will be three times the weight of the less cup. What is the weight of each cup? Ans. The less 3 oz. and greater 4 oz. 16. A saves the one-third of his wages; but B, who has, the same salary, by spending twice as much as A, sinks 52 dollars a year. What is their yearly salary? Ans. 156 dollars each. Suppose each bad % 108 A saves one-third 36 Errors 16-12=4)624 Aas. 156 dollars, salary of each. 00 |