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ca 4) Proof.

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. 1. How many changes may be made out of the three letters a, b, c, by taking two at a time?

CHANGES. a b 17 ba 21 ac 3.1

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cb 65 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 ?

PROBLEM III. Rollim 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. Eramelles 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 inan undertakes to throw an ace, at one throw, with 4 dice ; what is the probability of his effecting it?

Aos, as 671 to 625.

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MISCELLANEOUS QUESTIONS.

1. A gentleman bought 27 yards of cloth at 2s. per yard, 21 yards at 3s. 1 d. per yard, 25 yards at 1s. 8!d. 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 his 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 £2 175. 1d. ; 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 he step in a minute ?

Ans. 140%. 6. The divisor is 43967, the quotient 2737226, and the remainder 27672; what is the dividend ?

Ans. 120347613214. 7. A prize of 81000 is to be divided between two persons whose shares are in proportion of 7 to 9; required the share of each.

$437,50cts.

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8. After paying away and of my money, I had 66 guineas left in my purse ; what was in it at first ?

Ans. 120. 9. A reservoir 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.

10. In the latitude of Hallowell, a degree of longitude measures about 49 miles, 6 furlongs, and 114 poles; now as the earth turns round in about 23 hours, 56 minutes, at what rate per hour is the town of Hallowell carried by this motion from west to east ?* Ans. 748387 miles.

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 Ainerica, which lies under the equator, carried from west to east by this rotation ? Ans. 1045-45 miles.

12. In a mixture of wine cider, of the whole added to 25 gallons, was wine; and į part, less 5 gallons, was cider ; how many gallons were there of each ?

• Ans. 85 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 of the remainder is 80 ?

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 die vided among the sailors ; how much did each man receive i And The Capt. $1069,75cts.—The mate

nub $486,25cts. Each sailor $97,25 cents. 19. A stationer sold quills at 81,83 cts. per thousand, by which he cleared of the money; but as they grew scarcer, he raised them to 82,25cts. per thousand ; what did he clear per cent. by the latter price ?

Ans. $96,361cts. gained per cent. by the latter price.20. Bought a quantity of goods for $250, and 3 months

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after sold them for $275; how much per cent. per annum did I gain by them?

1. Ans. $40. 21. In what time will the interest of $72,60cts equal that of $15,25cts. for 64. days, at any rate of interest.?

Ans. 15461 days. 22. What sum of money. will amount to $132,81cts. 2}m. in 15 inonths, at 5 per cent. per annum simple interest ?

Ans. 8125.. 23. A person possessed of j of a ship sold of his share for 81260; what was the reputed value of the whole at the same rate ?.

Ans. 5040. 24. Of my of a farm I sell of ; what I then own is worth $185; What is that part, and what the value of the farm !

Aos. 737, and the farm $1200. 25. What number is that to which if of 8 of be added, the total will be one? . .Ans. 3646.

26. If of off of a ship be worth of of 4 of the cargo, valued at 40000 dollars; what is the value of the ship and cargo?:

Ans. $50744,81cts. + 27. A grocer would mix a quantity of sugar at 10d. per pound, with other sugars at 7d. 5d. and 4 d: per pound, intending to make up a compound worth 6d. per pound; what quantity of each must hé take? Ans. ill at 10d.

11 at 7jdi 13tb at 5d. and 410 at 43d. 28. If 1000 men besieged in a town, with provisions for 5 weeks, allowing each man 16 ounces a day, were reinforced with 500 men more, and hear that they cannot be relieved till the end of 8 weeks; how many ounces a day must each man have, that the provisions may last that time ? .

Ans. 6 ounces. 29. Sound, not interrupted, is found by experiment to move uniformly about 1150 feet in a second of time; how long then, after firing an alarm gun at Fort Independence, may the same be heard at Cambridge, taking the distance at 53 miles ?

Ans. 26 seconds. 30. If I see the flash of a gun fired by a vessel in distress at sea, which happens we will suppose at the instant of its going off, and hear the report a minute and 3 seconds afterwards; how far is she off? Ans.: 72450 feet.

31. An elm plank is 14 feet, 3 inches long; what distance from the edge must a line be struck to take off a yard square?

Ans. 71 inches.

32A man dying left his wife in expectation that a child would be afterwards added to the family, and in making his will ordered, that if the child were a son, of his estate should belong to him, and the remainder to his mother ; but if it were a daughter, he appointed the mother and the child the remainder ; but it happened that the addition was both a. son and a daughter, by which the widow lost in equity 2400 dollars more than if there had been only a girl; what would have been her dowry, had she had only a.son.?

Ans. $2100. 33. Having a piece of land 11 perches in breadth, I demand what length of it must be taken to contain an acre, when four perches in breadth require 40 perches in length to contain the samne ? Ans. 14per. 3yds.

34. If a gentleman whose annual income is €1000, spends 20 guineas, each 218., a week, will he fall in debt, or save money, and how much in the year ?

• Ans. £92 in debt. 35. What sum of money will produce as much interest in 34 years, as $210,15cts. can produce in 5 years and 5 monthsi

Ans. $350,25cts. 36. If 8100.in 5 years be allowed to gain 820, 50cts.. in what time will any sum of money double itself, at the same rate of interest?

Ans. 2419 years. 37. What difference is there between the interest of $350 at 4. per cent. for 8 years, and the discount of the same sum, at the same rate, and for the same time?

Ans. $27,15cts. 38. If by selling goods at $2 per cwt. I gain 20 per: cent., what do I gain or lose per cent. by selling at $24 per cwt. ?

Ans. $8 per cent. gain. 39. Required the length of a shore, which, strutting 11 feet from the upright of a building, may support a jamb 23 feet, 9 inches from the ground. Aos. 26ft. 2in. +

40. A clears $13 in 6 months, B $18 in 5, and C $23 in 9, his stock-being $72); what, then, is the general stock ?

Ans: $236,09 41. A person making his will, gave to one child 13 of his estate, and the rest to another; when these legacies came to be paid, the one turned out $600 more than the other : what did the testator die worth ? Ans. $2000.

42. A father devised of his estáté to one of his sons, and of the residue to another, and the remainder to his,

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