5. A Goldsmith sold 1 lb. of gold, at 2 cents for the furst ounce, 3 cents for the second, 52 cents for the third, &c. in a quadruple proportion geometrically; what did the whole come to? Hrs. $111848, 10cts. 6. What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthings, (or 24d.) the second, and so on, each month in a tenfold proportion ? Ans. £115740740 14s. 9d. 3qrs. 17. A thresher worked 20 days for a farmer, and receive ed for the first day's work four barley-corns, for the second 12 barley-corns, for the third 36 barley-corns, and so on in triple proportion geometrical. I demand what the 20 days?labor came to, supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel ? Ans. £1773 7s. 6d. rejecting remainders. 8. A man bought a horse, and by agreement was to give a farthing for the first nail, two for the second, four : for the third, &c. There were four shoes, and eight nails in each shoe ; what did the horse come to at that rate? Ans. £4473924 5s. 31d. 9. Suppose a certain body, put in motion, should move the length of one barley-corn the first second of time, one inch the second, and three inches the third second of time, and so continue to increase its motion in triple proportion geometrical ; how many yards would the said body move in the term of half a minute ? Ars. 953199685623 yds. ift. lin. 1b.c. which is no less than five hundred and forty-one millions of miles. 1 : POSITION. POSITION is a rule which, by false or supposed numbers, taken at pleasure, discovers the true ones required. It is divided into two parts, Single or Double. SINGLE POSITION, Is when one number is required, the properties of which are given in the question. RULE. 1. Take any number and perform the same operation with it, as is described to be performed in the question. 2. Then say; as the result of the operation : is to the given sum in the question ; : so is the supposed number : to the true one required. .' The method of proof is by substituting the answer in the question. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148; How many scholars had he ? Suppose he had 12 As 37 : 148 :: 12 : 48 Ans, as many = 1% . 16 12 Result, 37 Proof, 148 . 2. What number is that which being increased by }, }, and 7 of itself, the sum will be 125?... Ans. 60. 3. Divide 93. dollars between A, B and C, so that B's share may be half as much as A's, and C's share three times as much as B's. Ans. A's share $31, B's $15), and C's 846}. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3 times as much as A, and C took up as much as A and B both; what share of the gain had each 1 . Ans. A $40, B 8140, and C $180. :5. Delivered to a banker à certain sum of money, to receive interest for the same at 6l. per cent. per annum, simple interest, and at the end of twelve years received 7311. principal and interest together; what was the sum, delivered him at first ? Ans. £425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours and C in 4 hours; in what time will they all fill it together? Ans. 34min. 17 sec. hat DOUBLE POSITION, TEACHES to resolve questions by making two suppe sitions of false numbers. * RULE. 1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Find how much the results are different from the results in the question, 3. Multiply the first position by the last error, and the last position by the first error. .-4. If the errors are aliké, divide the difference of the products by the difference of the errors, and the quotient will be the answer. 5. If the errors are unlike, divide the sum of the prou ducts by the sum of the errors, and the quotient will be , the answer. NOTE: The errors are said to be alike when they are both too great, ou both too small; and unlike, when one is too great, and the other too small. EXAMPLES. 1. A purse of 100 dollars is to be divided among 4 men, A, B, C and D, so that B may have 4 dollars more than A, and C 8 dollars more than B, and D twice as many as C; what is each one's share of the money 1st. Suppose A 6 2. Suppose À 8 *Those questions, in which the results are not proportional to their positions, belong to this rule ; such as those in which the number sought is increased or diminished by some given number, which is no known part of the number grequired. ne 12 The errors being alike, are both too small, therefore, Pos. Err. 6 30 16 24 D 48 8 20: Proof 100 240. 1:20 120 110)120(12 A's part. %. A, B, and C, built a house which cost 500 dollars, of which A paid a certain sum; B paid 10 dollars more than A, and C paid as much as A and B both ; how much did each man pay? . Ans. A paid $120, B $150, and C $250. 3. A man bequeathed 1001. to three of his friends, after this manner: the first must have a certain portion, the second must have twice as much as the first, wanting sl. and the third must have three times as nuch as the first, wanting 151.; I demand how much each man must have ? Ans. The first £20 10s. second 6,33, third £46 10s. 4. A laborer was hired 60 days upon this condition ; that for every day he wrought he should receive 4s. and for every day he was idle should forfeit 25.; at the expiration of the time he received 71. 10s. ; how many days did he work, and how maty was he idle ? Ans. He wrought 45 days, and was idle 15 days. 5 . What number is that which being increased by its , its , and 18 more, will be doubled ? Ans. 72. 6. A man gave to his three sons all his estate in money, viz. to F half, wanting 501. to Gone-third, and to H the rest, which was 101. less than the share of G; I demand the sum given, and each man's part ? . Ans, the sun given was £360, whereof F had £130, G £120, and H £110. 7. Two men, A and B, lay out equal sums of money in trade; A gains 1261. and B looses 87l. and A's money is now double to B's; what did each lay out ?... Ans. £300, 8. A farmer having driven his cattle to market, receiv. ed for them all 1301. being paid for every ox 71. for every ców 5l. and for every calf 11. 10s. there were twice as many cows as oxen, and three times as many calves as COWS; how many were there of each sort? in Ans. 5 oxen, 10 cows, and 30 çalves. . 9. A, B and C, playing at cards, staked 324 crowns ; but disputing about tricks, each man took as many as he could : A got a certain number; B as many as A and 15 inore; C got a 5th part of both their sums added together; how many did each get :: Ans. Agot 1271, B 142), C 54. ...PERMUTATION OF QUANTITIES, Is the shewing how many different ways any given number of things may be changed. To find the number of Permutations or changes, that can be made of any given number of things, all different from each other RULE. : Multiply all the terms of the natural series of numbers, from one ip to the given nuinber, continually together, and the last product will be the answer required. EXAMPLES. 1. How many changes can be rilabc made of the three first letters of the alphabet ? Si Proof, 34 uca |