1. Then, X I A lady bought damask for a gown, at 8s. per yard, and lining for it, at 3s. per yard ; the gown and lining contained 15 yards, and the price of the whole was £3 10s.-How many yards were there of each ? Suppose 6 yards damak, value 48sa Then she must have 9 yards of lining, value 275. Sum of their values=755. So that the first error is 5 too much, or + 5 Again, suppose she had 4 yards of damask, value 325. Then she must have 11 yards of lining, value 335. Sum of their values=655. So that the second error is 5 too little, or 55. 6 5+ 5 yds. at 8s.=4,200 4 510 yds. at 35.= 100 30 £ 3 100 (Proof. Sum of errors. =5+5= 10)50 Anf. 5 yards damalk, and 15-5=10 yds. (lining Or, 6+4+2=5 as before. 2. A and B have the same income :-A faves of his ; but B, by spending £. 30 per annum more than A, at the end of 8 years finds himself £i 40 in debt; What is their income, and, what does each spend per annum? *30 Anf. their income is £ 200 Suppose Х (per annum. 160 40+ Also, A spends £175 and (B. 6205 per annum. *Then, 80-10= 70, A's expense per annum ; and 20 20 S*80 180+ annuim. firit error. 70+30 = 100, B's expense per Then 100 X 8 - 80 x 8=160, which fhould have been 40; therefore 160-40=120 more than it should be, for the In like manner proceed for the second error. 3. A and B laid out equal sums of money in trade : A gained a sum equal to of his stock, and B loft £ 225; then A's money was double that of B ; What did cach lay out ; $ 300 225+ Suppose X Anf. 4600. c2900 225– 4. A labourer was hired for 60 days upon this condition, that, for every day he wrought, he should receive 35. 4d. and for every day he was idle should forfeit is. 8d. at the expiration of the time he received £3 155. ; How many days did he work, and how many was he idle? 20 200Suppose he worked 240 300+ Anf. He was employed 35 days, and was idle 25. 5. A gentleman has two horfes of conderable value, and a carriage worth & 100. Now, if the first horse be harneffed in it, he and the carriage together will be triple the value of the fecond ; but if the second be put in, they will be 7 times the value of the first : What is the value of each horse ? 32 & X Anj. One £20, and the other £40. 6. There is a filh, whose head is to feet long ; his tail is as long as his head and half the length of his body. and his body as long as the head and tail; What is the whole length of the fish ? Head 10 ca'f ; there were 12 Head=10 First, fuppose the body 20 Tail=30 X Body=40 2d, fuppofe it 30 5 Anf. 80 feet. 7. What number is that, which, being increased by its i, its - , and 5 more, will be doubled ? 8 3+ Suppose X Anf. 20 216 16 it 8. A farmer having driven his cattle to market, received for them all, £80, being paid at the rate of £6 per ox, k4 per cow, and tos ios. per as many usen as cows, and 4 times as many calves as cows : How many were there of each fort ? Suppose cows 6 16+ X 112+ Anf. 5 oxen, 5 cows, and 20 calves. 9. A, B and C built a ship, which cost them £ 1000-of which A paid a certain fum-B paid £ 100 more than A, and C £100 more than both; having finished her, they fixed her for sea with a cargo worth twice the value of the ship : The outfits and charges of the voyage amounted to g of the ship ; upon the return of which, they found their clear gain to be } of į of the vessel, cargo and expenses : Please to inform me what the ship cost them, severally ; what share each had in her, and what, upon the final adjustment of their accompts, they had severally gained ? Suppose it cott A £100 300- X 100+ A owned io of the ship, which colt him £175, and his share of the gain was £218 155.--B owned 15, which cost £275, and his gaia was £ 343 155.C owned to which colt £550, and his gain was £,687 1os. PERMUTATIONS 200 PERMUTATIONS. and COMBINATIONS. The Permutation of Quantities is the fhewing how many different ways any given number of things may be changed. This is also called variation, alternation, or changes ; and the only thing to be regarded here is the order they stand in ; for no two parcels are to have all their quantities placed in the same fituation. The Combination of Quantities is the shewing how ofte « less number,of things can be taken out of a greater, and combined together, without considering their places, or the order they ftand in. This is sometimes called ekction, or choice ; and here every parcel must be different from all the rest, and no two are to have precisely the same quantities or things. The Composition of Quantities is the taking of a given number of quantities out of as many equal rows of dif. ferent quantities, one out of every row, and combining them together. Here, no regard is had to their places ; and it differs from Combination, only, as that admits but of one row of things. To find the number of permutations, or changes, that can be made of any given number of things, all different from each other. RuLF*- Multiply all the terms of the natural series of numbers, from 1 up to the given number, continually together, and the last product will be answer required. EXAMPLES * Any two things a and b are capable of two variations only; as, ab, ba ; whose number is expressed by 1X 2. If there be three things, a, b, and c, then any two of them, leaving out the third, will have 1x2 variations; and consequently when the third is taken in, there will be 1X2X3 variations; and so an, as far as you please. E S. I. 2. E XAMPL Christ Church, in Bostos, has 8 bells; How many changes may be rung on them ? 1 X2 X3 X4 X5 X6*78=40320 the Anf. 2. Nine gentlemen met at an inn, and were so pleased with their hoft, and with each other, that, in a frolic, they agreed to tarry so long as they, together with their hoit, could fit every day in a different position at Jinner; Pray how long, had they kept their agreement, would their frolic have lasted ? Anf: 994110 years. 3. How many changes or variations will the alphabet admit of? Anf. 620448401733239439360000. PROBLEM Any number of different things being given to find how many changes can be made out of them, by taking any given number of quantities at a time. RULE.--Take a series of numbers, beginning at the number of things given, and decreasing by 1, to the number of quantities to be taken at a time, the product of all the terms will be the answer required. EX A M P L E S. be 4 bells rout of 8? 8x7 x6x5(=4 terms)=1680 the An. How many words can be made with 6 letters of the alphabet, admitting a nunber of confonants may make a word? Anf. 96909120. PR ов и E м 3. Any number of things being given--whereof there are several things of one fort, feveral of another, &c. To find bow • many changes may be made out of them all. Rule *1.-Take the series 1 X 2 X 3 X 4, &c. up to the Any 2 quantities, a, b, bath different, admit of two changes ; but if the quantities are the same, or a., become aa, there will be only one alteration, which may be expressed by 1 X 2 Any 1. 2. |