of 40 oz. 18 carats fine; How much of each fort is required? Anf. 16 oz. 15 carats fine, 4 oz. 17, 8 oz. 20, and 12 oz. of 22 cărats fine? 3. How many gallons of water, of no value, muft be mixed with wine, at 4s. per gallo, fo as to fill a vessel of 80 gallons, that may be afforded at 23. 9d. per gallon ? 33 Gal. 15 Gal. Gal. Gal. 324833 As 49:80:: Sum 48 S15 25 Gallons of water? CAS E IV. When more than one of the fimples are limited. Anj 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 S. 1. How much wine at 4s. 6d. and at 5s. per gallon, must be mixed with 6 gallons at 4s. and 6 gallons at 35. per gallon, that the mixture may be worth 4s. 4d. per gal lon ? Now, having found the rate of the limited fimples, the queftion may ftand thus: How much wine, at 4. 6d. and 5s. per gallon, mult be mixed with 12 gallons, at 3s. 6d. per gallon, that the mixture may be worth 45. 4d. per gallon? Pofition is a rule, which, by falfe, or, fuppofed num. bers, taken at pleasure, discovers the true ones required. It is divided into two parts; Single and Double. SINGLE POSITION. Single Pofition teaches to refolve thofe queftions, whose results are proportional to their fuppofitions : Such are those which require the multiplication or divifion of the number fought by any propofed number; or when it is to be increased or diminished by itself a certain propofed number of times. RULE 1. Take any number, and perform the fame operations with it, as are described 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, and if it agrees with the fum, it is right. EXAM P LES. I. A fchoolmafter 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 num ber did his fchool confift? Suppofe Suppose he had 80. As 290: 435 :: 80 As many 80 2 as many=60 as many=40 80 120 29|0)34800(120 Anf. 120 2. A perfon lent his friend a fum of money unknown, to receive intereft for the fame at 67. per cent. per annum, fimple intereft, and at the end of 12 years, received for principal and intereft 860/.; What was the fum lent? Anf. 500l. 3. A, B and C joined their ftocks, and gained 350%; 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 share of the gain had each ? £. s. 9 9 37 16 d. qrs. 1 A's fhare. 1 2 22 302 14 o 2 C's ditto. 4. A, B, C and D spent 35s. at a reckoning, and, being a little dipped, they agreed that A fhould pay 4, B, C, and D ; What did each pay in the above proportion? s.. d. A, 13 4 Anf √ B, 10 C, 68 D, 5 5. A certain fum of money is to be divided between 5 men, in fuch a manner as that A fhal have, B, C To, D, and E the remainder, which is is the fum ? 40; What Suppofe £200 then +++26=120. 200-120 = 80, As 80 : 40 :: 200: 100, Anf. 6. A perfon, after spending and had 26 left; What had he at firft? 7. of his money, Anf. £160. 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? 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? of an Anf. hour. 9. A perfon having about him a certain number of dollars, faid that 4, 4, 4, and of them would make 57; Pray, how many had he? for io. Anf. 60. A gentleman bought a chaife, horfe and harness 100; the horfe coft more than the harness, and the chaife more than the horfe; What was the price of each ? of Anf Har. £254, horfe £3144, chaife £4249. II. A and B. having found a purfe of money, difputed who fhould have it: A faid that, and it amounted to £35, and if B could tell him how much was in it he should have the whole, otherwife he should have nothing: How much did the purfe contain? Anf. £100. 12. A gentleman divided his fortune among his fons; to A he gave £9 as often as to B £5; and to C but as often as to B £7, yet C's portion came to 10503; What was the whole eftate? Anf. £791635 13. Seven eighths of a certain number exceeds four fifths by 6; What is that number ? 2 Anf. 80. 14. What number is that, which, being increased by , and of itfelf, the fum will be 234? 96 Anf. 90. DOUBLE DOUBLE POSITION. Double Pofition teaches to refolve queftions by making two fuppofitions of falfe numbers. Thofe queftions, in which the refults are not propor tional to their pofitions, belong to this rule; fuch are thofe, in which the number fought is increafed or diminithed 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 queftion. 2. Place the refult or errors against their pofitions or fuppofed Pos. Err. mark it with + ; and if too fmall with 3. Multiply them croffwife; that is, the firft pofition by the laft 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 anfwer. 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 first error is to the fecond, as the difference between the true and first fuppofed number is to the difference between the true and fecond fuppofed number; when that is not the cafe, the exact answer to the queftion cannot be found by this rule. |