of 40 oz. 18 carats fine ; How much of each fort is required? Ans. 16 oz. 15 carats fine, 4 0%. 17, 8 0%. 20, and 12 0%. of 22 cảrats fine? 3. How many gallons of water, of no value,' must be mixed with wini, at 45. per gallo , so as to fill a vessel of 80 gallons, that may be afforded at 25. 9d. per gallon? Gal. Gul. Gal. Gal. } Anji Sum 48 When more than one of the fimples are limited. RULE.Find, by Alligation Medial, what will be the rate of a mixture made of the given quantities of the limited simples only'; then, consider this as the rate of a limited fimple, whose quantity is the sum of the first given linited fimples, from which and the rates of the unlimited fimples, by Cafe 2d, calculate the quantity. Gal. s. 1. How much wine at 45. 6d. and at 5s. per gallon, must be mixed with 6 gallons at 45. and 6 gallons at 35. per gallon, that the mixture may be worth 4s. 4d. per gale lon? Gal. Limited S6 Gallons at 45.=24} As 12:42 :: 1 : 3/6 fimples. 16 Gallons at 38. =18 (per gallon. 42 Now, having found the rate of the limited simples, the question may stand thus : How much wine, at 45. 6d. and $s. per gallon, mult be mixed with 12 gallons, at 3s. 6d. per gallon, that the mixture may be worth 45. 4d. per I2 gallon ? Position is a rule, which, by falle, or fupposed numbers, taken at pleasuri, discovers the true ones required. It is divided into two parts ; Single and Double, SINGLE POSITION. Single Position teaches to resolve those questions, whose results are proportional to their fuppofitions : Such are those which require the multiplication or divi. fion of the number fought by any proposed number ; or when it is to be increased or diminished by itself a certain proposed number of times. Rule 1.--Take any number, and perform the same operations with it, as are described to be performed in the question. Then say; As the sum of the errors is to the given sum ; fo is the supposed number to the true one required. Proof.—Add the several parts of the sum together, and if it agrees with the fum, it is right. 2. A schoolmafter being asked how many fcholars he had, faid, If I had as many more as I now have, three quarters as many, half as many, one fourth and one eighth as many, I should then have 435; Of what numă ber did his school confift? Suppose Suppose he had 80. As 290 : 435 :: 80 80 120 as many=40 2910)34800(120 Anf. 120 as many=20 29 ୨O as many10 60 *30 290 15 58 58 2. 435 Proof. A person lent his friend a sum of money -unknown, to receive interest for the same at 61. per cent. per annum, fimple interest, and at the end of 12 years, received for principal and interest 8601.; What was the fum lent ? Anf. 5ool. 3. A, B and C joined their stocks, and gained 350!. ; of which, A took up a certain fum ; B took up four times so much as A, and C, eight times so much as B; What share of the gain had each ? £ d. qrs. 12 C's ditto. 4. A, B, C and D spent 355. at a reckoning, and, being a little dipped, they agreed that A should pay }, B, C }, and D*;. What did each pay in the above proportion ? d. 2 { SA; 13 4 A, 13 4 28 Ans. C, 68 D, 5 5. A certain sum of money is to be divided between 5 men, in fuch a manner as that A shah have , BC 15, D 36, and E the remainder, which is £ 40 ; What is the fum ? Suppose to 200 then ++ +to+=120. 200-120 = 80, As 80 : 40 :: 200 : 100, : Ans. 6. A person, after spending { and of his money, had £261 left; What had he at first? Anf. £160. 7. A and B talking of their ages, B said his age was once and a half the age of A ; C said his was twice and one tenth the age of both, and that the sum of their ages was 93-; What was the age of each? Ans. A's 12, B's 18, and C's 63 years, 8. A vessel has 3 cocks, A, B and C; A can fill it in half an hour, B in of an hour, and C in į of an hour; In what time will they all'fill it together?... Anfo: hour. 9. A person having about him a certain number of dollars, said that , , š, and ; of them would make 57; Pray, how many Anf. 6o. io. A gentleman bought a chaise, horse and harness for £100 ; the horfe coft more than the harness, and the chaise } more than the horse ; What was the price of each? Anf. Har. £2574, horse £314!, chaise £1275. u. A and B having found a purse of money, dil. puted who should have it : A said that fy to and to of it amounted to £ 35, and if B could tell him how much waz in it he should have the whole, otherwise he should Lase nothing: How much did the purse contain ? Any: £100. A gentleman divided his fortune among his sons ; to A he gave £9 as often as to B £5; and to Ç but £3 as often as to B £7, yet C's portion came to 1050f ; What was the whole eitate ? Ans: £7916754 13. Seven eighths of a certain number exceeds four fifths by 6;. What is that number? Anf. 80. 14. What number is that, which, being increased by }, and of itself, the fun will be 234.1? 12. Ans: 90. DOUBLE DOUBLE POSITION 9 12 Double Position teaches to resolve questions by making two suppositions of falfe numbers. Those questions, in which the results are not propor. tional to their positions, belong to this rule ; such are those, in which the number fought is increased or diminished by some given number which is no known part of the number required. Rule* 1.--Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Place the result or errors against their positions or supposed Pos. Err. 30 numbers, thus, X and if the error be too great 6 mark it with + ; and if too small with 3. Multiply them crosswise ; that is, the first pofie tion by the last error, and the last position by the first 4. If the errors be alike ; that is, both too small, or both too great, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. If the errors be unlike ; that is, one too small, and the other too great, divide the sum of the products by the sum of the errors, and the quotient will be the anfwer. Note, When the errors are the same in quantity, and unlike in quality, half the sum of the fuppofitions is the number fought. 20 error. EXAMPLES * The rule is founded on this fuppofition, that the first error is to the second, as the difference between the true and first fuppofed number is to the difference between the true and second supposed number; when that is not the case, the exact answer to the queftion cannot be found by this rule. |