POSITION, or the RULE of FALS. POSITION, fuppofition or the rule of falfe is a rule wherein falfe or supposed numbers are used, in order to discover true ones; and is applied to the performing of fuch questions as cannot be refolved by the common direct rules. SINGLE POSITION. SINGLE pofition teacheth how to perform those queftions that require but one fuppofition to difcover the true refult. R U L E. Suppofe any number, and perform the fame operations with it as are described to be performed in the queftion. Then ftate, as the refult of the operation is to the fuppofed number: fo is the refult in the queftion to the number fought. NOTE. The number 1 may be made a conftant fuppofition in all queftions; and, in moft cafes, it is better than any other number. PROOF. Add the feveral parts of the fum together, and if the fum agrees with the given number, the operation is right. 1. What number is that to which if we add the half, the fourth, and the fifth of itself the whole fum. will be 39? Suppose the number to be 40 As 78 40 39 2. Find a number which being multiplied by 8, and the product divided by 5, the quotient will be 24. Anf. 15 3. Two geese, 4 ducks, and 8 hens cost together 14 fhillings. A goofe coft twice the price of a duck, and 5 times that of a hen; what did each coft? Anf. A goofe coft 2s. 6d. a duck 1s. 3d. a hen 6d. 4. A perfon bought a chaife, horfe, and harnefs for6ol. The horse came to twice the price of the harness, and the chaise to twice the price of the horfe and harnefs. What did he give for each ?-Anf. 131. 6s. 8d. for the horse, 61. 135. 4d. for the harness, and 40l. for the chaife. 5. Required a number the fquare whereof is equal to 4 times the number itself? Anf. 4. 6. A man carrying a bag of money in his hand, was asked how much was in it. He answered that the third, fourth, and fifth of it made 941. How much money had he? Anf. 120l. 7. What number is that which being increased by and of itself shall be 63 ? Anf. 36. 8. A general being afked the number of m army confifted of, anfwered, that of amounte 900; what number of men had he? Anf.. 2400 me. 9. A man being asked his age, faid, if 3 of the year. I have lived be multiplied by 7, and of them be added to the product, the fum will be 292; what was his age? Anf. 60 years. REMARK. All the queftions in pofition whofe refults are proportional with their fuppofitions, or with fome power or root of their fuppofitions, may be refolved by the above rule. Such as are not in one of the abovenamed fituations belong to double pofition. DOUBLE POSITION. DOUBLE pofition teacheth how to perform thofe queftions that require two fuppofitions to find the number fought. R U L E. Take any two convenient numbers, and proceed with each according to the conditions of the queftion. Then, find how much the refults are different from the refults in the question, and call thefe differences the errors. This done, multiply each of the errors by the contrary fuppofition, and find the fum or difference of the products; and then, if the errors are alike, divide the difference of the products by the difference of the errors for the anfwer; but if they are unlike, the quotient of the fum of the products divided by the fum of the errors will be the answer. NO T E. rft. The errors are faid to be alike, when the refults are both too great or both too little; but when one is too great and the other too little, they are faid to be unlike. zd. It will be often advantageous to make 1 and o the fuppofitions. EXAMPLE S. 1. John's age is now double Thomas's; but 15 years ago it was three times Thomas's; what are their ages. now? yrso Suppofe John's age to be 80, then Thomas's will be 40 From which fubtract 15 15 25 3 75 Suppofe John's age to be 70, then Thomas's will be 35 yrs. 15 20 3 60 80=400 53.00 John's age 60. An£ yrs. John's age is 60, and Thomas's will be 30 2. A man left his 3 fons his eftate in money; to F half wanting 50l.; to G one-third; and to H reft, which was 1ol. lefs than the share of G; I deman each man's part?-Anf. F's fhare was 130l. G's 120l. and H's Iol. 3. Double my money for me, faid John to Tom, and I will give thee 6d. out of the stock. With the remainder he applied in like manner to Dick, with equal fuccefs, and gave him alfo 6d. He repeated this propofal to Ned, and then 6d. was all he had to give. Pray what fum had he to begin with? Anf. 54d. 4. A labourer was hired for 40 days, upon this condition, that he fhould receive 20d. for every day he wrought, and forfeit 10d. for every day he was idle. Now he received at last 21. 15. 8d.; how many days did he work, and how many was he idle?-Anf. He wrought 30 days, and was idle 10 days. 5. A perfon gave 141. 8s. for a hogfhead of wine, which was a mixture of wine at 4s. and at 5s. the gallon. Required the quantity of each fort?-Anf. 27 gs. at 4s. and 36 gs. at 58. 6. There is a fifh whofe head is 9 inches long, and his tail is as long as his head and half as long as his body, and his body is as long as his tail and his head; what is the whole length of the fish? Anf. 3 feet. 3 7. Old John, who had in credit liv'd, Than clam'rous duns came fwarming round. Anf. 1411. |