40oz. 18 carats fine; how much of each fort is r quired? Anf. 16oz. 15 carats fine, 4oz. 17, 8oz. 20, and 12oz. of 22 carats fine. 3. How many gallons of water, of no value, muft be mixed with wine at 4s per gallon, fo as to fill a veffel of 80 gallons, that may be afforded at 2s. 9d. per gallon. 33 Gal. 48 33 Gal. Gal. Gal. Sum 48 As 48: 80 :: 484 S15:25 of water CASE IV. } 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 fimples only; then, confider this as the rate of a limited fimple, whofe quantity is the fum of the first given limited fimples, from which and the rates of the unlimited fimples, by Cafe 2d. calculate the quantity. EXAMPLE. How much wine at 4s. 6d. and at 5s. per gallon, must be mixed with 6 gallons at 4s. and 6 gallons at 3s. per gallon, that the mixture may be worth 4s. 4d. per gallon? Gal. s. Gal. Limited 56 gallons at 4s. 24 As 12: 42 :: 1 : 3⁄6 fimples 6 gallons at 3s.18 S (per gal. Now, having found the rate of the limited fumples, the queftion may ftand thus: How much wine at 4s 6d. and 5s. per gallon, must be mixed with 12 gallons at 4s. 6d. per gallon, that the mixture may be worth 4s. 3d. per gallon? 52 Pofition is a rule, which by falfe, or fuppofed numbers, taken at pleasure, difcovers the true ones required. It is divided into two parts; Single and Double. SINGLE POSITION. Single Pofition teaches to refolve thofe questions, whofe refults are proportional to their fuppofitions ; fuch as those which require the multiplication or divifion of the number fought by any propofed number; or when it is to be increafed or diminished by itself a certain propofed number of times. RULE 1. Take any number and perform the fame operations with it, as are defcribed to be performed in the question. 2. Then fay; as the fum of the errors is to the given fum; fo is the fuppofed number to the true one required. PROOF. Add the feveral parts of the fum together agrees with the fum it is right. and if it EXAMPLES. A schoolmafter being asked how many scholars 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 fhould then have 435; of what number did his fchool confift? Suppofe Suppofe he had 80 As 290: 435 :: 80 2. as many = 60 80 as many = 40 29|0)34800(120 Anf. 120 120 90 60 30 15 Proof 435 A perfon lent his friend a fum of money unknown, to receive interest for the same at 61. per cent. per annum, fimple intereft, and at the end of 12 years, received for principal and interest 8601.; what was the fum lent? Anf 500l. 3. A, B and C joined their stocks, and gained 350l.; of which A took up a certain fum; B took up four times fo much as A, and C eight times fo much as B ; what fhare of the gain had each ? £. s. d. qrs. 2 3 37 A's fhare. 90 B's ditto. 302 14 0 234 C's ditto. 4. A, B, C and D fpent 35s. 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? 5. A certain fum of money is to be divided between 5 men in fuch a manner as that A fhall have 4, B, C Yo, Do, and E the remainder, which is 40l. ; what is the fum ? Suppofe £200 then +++120. 200-120=80, As, 80: 40 :: 200: 100, Anf. 6. A perfon after spending and of his money, - lad 2631 left; what had he at first? Anf £160. 7. A and B talking of their ages, B faid his age was once and a half the age of A; C faid his was twice and one tenth the age of both, and that the fum of their ages was 93; what was the age of each? of an Anf. A's 12, B's 18 and C's 63 years. 8. A veffel has 3 cocks, A, B and C ; A can fill it in half an hour. B in of an hour and C in hour; in what time will they all fill it together? An hour. 9. A perfon having about him a certain number of dollars, faid that, 4, and of them would make 57; Pray how many had he? Anf. 60. 10. A gentleman bought a chaife, horfe and harness for tool. ; the horse coft more than the harness, and the chaife more than the horse; what was the price of each ? 26 Anf Har. £25, horfe £3143, chaife £4229. 11. A and B, having found a purse of money, difputed who fhould have it; A faid the and of it amounted to 351. and if B could tell how much was in it he should have the whole, otherwise he should have nothing: how much did the purse contain? Anf. 100l. 12. A gentleman divided his fortune among his fons ; to A he gave las often as to B 51.; and to C but 31. as often as to B 71. yet C's portion came to 10503; what was the whole eftate? Anf £9716323. 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 1 ", and of itself, the fum will be 2344? DOUBLE Anf. 90. DOUBLE POSITION. Double Pofition teaches to refolve questions by mak ing two fuppofitions of falfe numbers. Thole queftions, in which the refults are not proportional to their pofitions, belong to this rule; fuch are thofe in which the number fought is increased or diminished by fome 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 refult or errors against their positions of fuppofed Pos. Err. 30 12 numbers thus, X 20 6 and if the error be too great mark it with+; and if too fmall with 3. Multiply them croffwife; that is, the first pofition by the last error, and the laft pofition by the first error. 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 anfwer. 5. If the errors be unlike; that is, one too small, and the other too great, divide the fum of the products by the fum of the errors, and the quotient will be the an'fwer. Note, When the errors are the fame in quantity, and unlike in quality, half the fum of the fuppofitions is the number fought. EXAMPLES. *The rule is founded on this fuppofition, that the firfterror is to the fecond, as the difference between the true and firft fuppofed number is to the difference between the true and fecond fuppofed hamber; when that not the cafe, the exact answer to the queftion cannot be found by this rule. |