{ A, at the end of 4 years is 100 dollars in debt ; what is their income, and what do they spend per annum ? Their income is $125 per year. Ans. A spends $100. B spends $150. 3. A labourer was hired for 40 days upon these conditions, that he should receive 2 dollars for every day he wrought, and forfeit 1 dollar for every day he was idle ; at the expiration of the time he was entitled to 50 dollars ; how many days did he work, and how many was he idle ; Ans. He wrought 30 days, and was idle 10. 4. A man had 2 silver cups of unequal weight, with i cover for both, weight 5oz; now if he put the cover on the less cup, it will be double the weight of the greater; and put on the greater cup, it will be three times the weight of the less cup; what is the weight of each cup? Ans. 3oz, the less, and 4oz. the greater. 5. A person being asked what o'clock it was, answered that the time past from noon was equal to is of the time to midnight ; required the time. Ans. 36 minutes past 1. 6. There is a fish whose head is ten feet long ; bis tail is as long as his head and half the length of his body, and his body is as long as his head and tail ; what is the whole length of the fish ? Ans. 80 feet. 7. A and B laid out equal sums of money in trade; A gained a sum equal to 1 of his stock, and B lost 225 dollars; then A's money was double that of B's; what did each lay out. Ans. $600. PERMUTATION AND COMBINATION. THE permutation of quantities is the showing how many different ways the order or position of any given number of things may be changed. The combination of quantities is the showing how often a less number of things may be taken out of a greater, and combined together, without considering their places, or the order in which they stand. R PROBLEM 1. To find the number of permutations, or changes, that can be made of any number of things, all differing from each other. RULE.—Multiply all the terms of the natural series of numbers, from one up to the given number, continually together, and the last product will be the answer. CHANGES. EXAMPLES. 1. How many changes may be made with these three letters, A, B, C. 1 a b c 17 acb 2 Proof. b c a 41 cb a 65 Ans. 720. 3. How many changes may be rung upon 12 bells, and how long would they be ringing but once over, supposing 10 changes might be rung in one minute, and that the year contains 365 days, 6 hours ? 479001600 changes, and 91 Ans. years 3w. 5d. and 6 hours. 4. A young scholar coming into a town for the convenience of a good library, demanded of the gentleman with whom he lodged, what his diet would cost for a year ; he told him $150; but the scholar not being certain what time he should stay, asked him what he should give him for so long as he could place his family (consisting of 6 persons beside himself) in different positions every day at dinner; the gentleman told him 850 ; to this the scholar agreed—what time did he stay ! Ans. 5040 days. PROBLEM II. Any number of different things being given, to find how many changes can be made out of them, by taking any given number at a time. Rule.-Take a series of numbers, beginning at the number of things given, and decreasing by one, till the number of terms be equal to the number of things to be taken at a time, multiply these terms into each other; and the product will be the answer. EXAMPLES. ас 1. How many changes may be made out of the three letters a, b, c by taking two at a time? CHANGES. 3 a b 1 1 b a 2 Proof. са 4 cb6 2. How many changes may be made with the nine digits, by taking 3 at a time? Ans. 504. 3. How many words may be made with the alphabet by taking five letters at a time, supposing that a number of consonants may make a word ? Ans. 5100480. PROBLEM III. To find the compositions of any number, in an equal number of sets, the things themselves being all different. Rule.--Multiply the number of things in every set continually together, and the product will be the answer. EXAMPLES. 1. Suppose there are four companies, in each of which there are nine men; it is required to find how many ways four men may be chosen, one out of each company. 9x9x9x9=6561 the answer. 2. How many changes are there in throwing five dice ? Ans. 7776. 3. Suppose there are four companies, in one of which there are 6 men, in another 8, and in each of the other two 9; what are the choices by a composition of four men, one out of each company ? Ans. 3888. 4. Suppose a man undertakes to throw an ace, at one throw, with 4 dice ; what is the probability of his effecting it? Ans. as 671 to 625, MISCELLANEOUS QUESTIONS. a а : 1. A gentleman bought 27 yards of cloth at 2s. per yard, 24 yards at 3s. 1 d. per yard, 25 yards at 1s. 84d. per yard; he also bought 3 yards of broadcloth, the price of which he does not recollect; but on counting his money he found he had expended &11 19s. 2 d. ; what did bis broadcloth cost per yard, in Federal Money! Ans. $3,75cts. 2. A servant went to market with £5, and bought eggs at 7 for 4d. ; 2 pair of fowls at 2s. 4d. a pair; 17 pigeons at 3s. per dozen ; 3 rabbits at 14d. each; and 3 dozen of larks at 14d. per dozen; he also paid the baker £2 17s. id. ; when he returned he had 21s. left; how many eggs did he buy? Ans. 126. 3. I have a drawer 17 inches long, 12 inches broad and 7 inches deep ; how many one inch dice will it hold ? Ans. 1428. 4. At a certain election 375 persons voted, and the candidate chosen had a majority of 91 : how many voted for each ? Ans. 233 and 142. 5. Suppose a man to step 30 inches at a time, and to go 4 miles an hour; how many times does be step in a . minute ? Ans. 1404 6. The divisor is 43967, the quotient 2737226, and the > remainder 27672; what is the dividend ? Ans. 120347643214. 7. A prize of $1000 is to be divided between two persons whose shares are in proportion of 7 to 9; required the share of each. . . 8. After paying away i and } of my money, I had 66 guineas left in my purse ; what was in it at first ? Ans. 120. 9. A reserveír for water has two cocks to supply it ; the first alone will fill it in 40 minutes, the second in 50 minutes and it has a discharging cock by which it may be emptied, when full, in 25 minutes. Now supposing that these three cocks are all opened, that the water comes in, and that the influx and the efflux of the water are always alike, in what time would the cistern be filled ? Ans. 3 hours, 20 minutes, Ans. { 1487,50cts. 10. In the latitude of Hallowell, a degree of longitude measures about 49 miles, 6 furlongs, and 11} poles ; now as the earth turns 'round in about 23 hours, 56 minutes, at what rate per hour is the town of Halowell carried by this motion from west to east ?* Ans. 74839miles. 11. If the earth turns round in 23 hours 56 minutes, at what rate per hour are the inhabitants of the city of Quito in South America, which lies under the equator, carried from west to east by this rotation ? Ans. 1045145 miles. 12. In a mixture of wine cider, of the whole added to 25 gallons, was wine ; and } part, less 5 gallons, was cider ; how many gallons were there of each ? Ans. 35 of wine and 35 of cider. 13. A hare is 50 of her own leaps before a greyhound, and takes 4 leaps to the greyhound's 3 ; but two of the greyhound's leaps are as much as 3 of the hare's : how many leaps must the greyhound take to catch the hare ? Ans. 300. 14. Out of a cask of wine which had leaked away part, 21 gallons were drawn; and then being gauged, it was found to be half full; how many gallons did it hold ! Ans. 126. 15. What part of 4d. is of 6 pence ? Ans.. 16. What number is that from which, if 5 be subtracted ß of the remainder is 30 ? Ans. 125. 17. A post is in the mud, } in the water, and 10 feet above the water : what is its whole length ? Ans. 24. 18. A captain, mate, and 20 seamen, took a prize worth $3501 ; of which the captain takes 11 shares, and the mate 5 shares; the remainder of the prize is equally divided among the sailors ; how much did each man receive ? Ans. { The Capt. $1069,75cts.—The mate 19. A stationer sold quills at $1, 83 cts. per thousand, by which he cleared 3 of the money ; but as they grew scarcer, he raised them to $2,25cts. per thousand ;what did he clear per cent. by the latter price ? Ans. $96,36 Acts. gained per cent. by the latter price. 20. Bought a quantity of goods for $250, and 3 months { $ * The earth moves 1 degree in 36 59" 204f of side. real time, or in 4' of solar or tropical time |