Problem 9. Having the circumference to find the side of a square which may be inscribed. If the circumference be 1, the side of the square inscribed will be,2251. Therefore, as 1 :,2251 :: 71 : 15,9821, the side of the square. Problem 10. Having the area, to find the diameter. If the area of a circle be 1, the square of the diameter is 1,2723. Therefore, as 1: 1,2732 :: 401,15: 510,744180, the square root of which is 22.599, the diameter. Problem 11. Having the area, to find the circumference, If the area of a circle be 1, the square of the circumference will be 12,56637. Therefore, as 1:12,56637 :: 401,15: 5040 ,99932550, the square root of which is 70,9999 Problem 12. Having the area, to find the side of a square scribed. in If the area of a circle be 1, the area of a square inscribed within that circle will be ,6366. Therefore, as 1 : 401,15 :: ,6366: 255,372090, the root of which is 15,980, the side of the square sought. Problem 13. Having the side of a square, to find the diameter of the circumscribing circle. If the side of a square be 1, the diameter of a circle that will circumscribe that square, will be 1,4142. Therefore, as 1: 1,4142: 15,98: 22,598916, the diameter sought. Problem 14. Having the side of a square to find the diameter of a square equal to it. If the side of a square be 1, the diameter of a circle equal to it will be 1,128. Therefore, as 1 : 1,128 :: 20,0291: 22,5928248 the diameter required. Problem 15. Having the side of a square, to find the cireumference of a circumscribing circle. If the side of a square be 1, the circumference of a circle that will encompass that square will be 4,443. Therefore, as 1: 4,443 15,98: 70,99914, the circumference required. Problem 16. Having the side of a square, to find the cir cumference of a circle that will be equal to it If the side of a square be 1, the circumference of a circle that will be equal to it is 3,545. Then, as 1: 3,545 :: 20,0291 : 71,0031595, the circumference. Note. In several of the foregoing problems, where the diameter and circumference are required, the answers are not exactly the same as the diameter and circumference of the given circle, but are sometimes too much, and sometimes too little, has in the two last problems, where the answers in each should be 71, the one being too much, and the other too little. The reason of this is, the small defect that happens to be in the decimal fractions, they being sometimes too great, and sometimes too little; yet the defect is so small, that it is needless to calculate them to more exactness. Of the Semicircle. To find the are of a semicircle, multiply the fourth part of the circumference of the whole circle by the semi-diameter, and the product will be the area. Suppose the diameter be 22,6 and the half circumference, or arch line, is 35,5. The half of it is 17.75, which multiplied by the semi-diameter 11,3, the product is 200,575, the area of the semi-circle. Of the Quadrant. To find the area of a quadrant, or the fourth part of a circle, multiply half the arch line of the quadrant, that is, the eighth part of the circumference of the whole circle (by the semi-diameter) and the product will be the area of the quadrant. These are the rules commorly given for finding the area of a semi-circle and quadrant; or find the area of the whole circle, and then take half the area for the semi-circle, and the fourth part for the quadrant. To find the Solidity of a Sphere or Globe. A sphere or globe is a round solid body, every part of its sur face being equally distant from a point within it, called its centre. To find its solidity, multiply the axis, or diameter, into the circumference, the product of which is the superficial conThis multiplied by a sixth part of the axis, the product is the solidity. tent. WINDOW DUTY. 48 GEO. III. Chap. 55.` For every Dwelling-House, not exceeding six windows, and under the annual rent of £5 per anuum, 6s. 6d. And for every Dwelling-House worth £5 per annum, and charged with the duty of an Inhabited House, 8s. THE TUTOR'S ASSISTANT. PART VI. A Collection of Questions. 1. WHAT is the value of 14 barrels of soap at 41d. per lb. each barrel containing 254 lb.? Ans. £66...13...6. 2. A and B trade together: A puts in £320 for 5 months, B £460 for 3 months, and they gained £100. What must each man receive? Ans. A £53...13...9270, and B £46...6...2. 3. How many yards of cloth, at 17s. 6d. per yard, can I have for 13 cwt. 2 grs. of wool at 14d. per lb. ? Ans. 100 yards, 3 qrs. z. 4. If I buy 1000 ells of Flemish linen for £90, what may I sell it per ell in London, to gain £10 by the whole? Ans. 3s. 4d. per ell. 5. A bas 648 yards of cloth, at 14s. per yard, ready money, but in barter will have 16s. B has wine at £42 per tun, ready money the question is, how much wine must be given for the cloth, and what is the price of a tun of wine in barter? Ans. £48 the tun, and 10 tun, 3 hhds. 123 gal. of wine must be given for the cloth. 6. A jeweller sold jewels to the value of £1200, for which he received in part 876 French pistoles, at 16s. 64. cach, what sum remains unpaid? Ans. £477...6. 7. An oilman bought 417 cwt. 1 qr. 15 i. gross weight of train oil, tare 20 lb. per 112 lb. how many neat gallons were there, allowing 74 to a gallon? Ans. 5120 gallons. 174 8. If I buy a yard of cloth for 14s. 6d. and sell it for 16s. 9d. what do I gain per cent.? Ans. £15...10...4 24 9. Bought 97 bags of ginger, each weighing gross 843 lb. tare 18 lb. per bag, tret 4 lb. per 104 lb. what do they come to at sid. per lb. ? Ans. £76...13...2. 10. If of an ounce cost of a shilling, what will of a lo. cost? Ans. 17s. 6d. of a £. what will of a ton cost? Ans. £105. 11. If of a gallon cost 12. A gentleman spends one day with another £1...7...104. and at the year's end layeth up £340, what is his yearly income? Ans. £848...14...4. 13. A has 13 fother of lead to send abroad, each being 19 times 112. B has 39 casks of tin, each 388 lb. how many ounces difference is there in the weight of these commodities? Ans. 2121C0 of. 4. A captain and 160 sailors took a prize worth £1360, of which the captain had for his share, and the rest was equally divided among the sailors, what was each man's part? Ans. the captain had £272. and cach sailor £6...16. 15. At what rate per cent. will £956 amount to £1314...10. in 7 years, at simple interest? Ans. 5 per cent. 16. A hath 24 cows worth 72s. each, and B 7 horses worth £13 a piece, how much will make good the difference, in case they interchange their said drove of cattle? Ans. £4...12. 17. A man dies and leaves £120 to be given to three persons, vix. A, B, C: to A a share unknown; B twice as much as A, and C as much as A and B; what was the share of each? Ans. A £20, B £40, and C £60. 18. There is a sum of £1000 to be divided among 3 men, in such a manner, that if A has £3, B shall have £5, and C £8, how much must each man have? Ans. A £187...10, B £312...10, and C £500. 19. A piece of wainscot is 8 feet 6 inches and long, and 2 feet 9 inches broad, what is the superficial content? Aus. 24 feet 0...3...4...6. 20. If 360 men be in garrison, and have provisions for 6 months, but hearing of no relief at the end of 5 months, how many men must depart, that the provisions may last so much the longer? Ans. 288 men. 21. The less of two numbers is 187, their difference is 34, the square of the product is required? Ans. 1707920929. 22. A butcher sends his man with £216 to a fair to buy cattle; oxen at £11, cows at 40s. colts at £1...5, and hogs at £1...15 per piece, and of each a like number, how many of each sort 'did he buy? Ans. 13 of each sort, and 8 over. 13. What number, added to 119 will produce 3337 ? Ans. 241. 24. What number, multiplied by 3, will produce 11? Ans. 261. 25. What is the value of 179 hogsheads of tobacco, each weighing 13 cwt. at £2...7...1 per cwt.? Ans. £5478...2...11. 26. My factor sends me word he has bought goods to the value of £500...13...6 upon my account, what will his commission come to at 3 per cent.? Ans. £17...10...5.2 qrs. 68 27. If of 6 be three, what will † of 20 be? Ans. 7. Ans. ,875 29. How many lb. of sugar, at 44. per lb. must be given in barter for 60 gross of incle, at 8s. 8d. per gross? Ans. 1386. 50 If I buy yarn for 9d. the lb. and sell it again for 134d. per lb. what is the gain per cent. ? Ans. £50. 31 A tobacconist would mix 20 lb. of tobaco at 9d. per lb. with 60 lb. at 12d. per lb. 40 lb. at 18d. per lb. and with 12 B. at 2s. per lb. what is a lb. of this mixture worth? Ans. 1s. 2Įd.{{ 32. What is the difference between twice eight and twenty and twice twenty-eight: as also between twice five and fifty and twice fifty-five? Ans. 20 and 50. 33. Whereas a noble and a mark just 15 yards did buy ; How many ells of the same cloth for £50 had I? Ans, 600. 34. A broker bought for his principal in the year 1720, £400 capital stock in the South Sea, at £650 per cent. and sold it again when it was worth but £130 per cent. how much was lost in the whole? Ans. £2080. 35. Chath candles at 6s. per dozen ready money, but in barter will have 6s. Ed. per dozen; A hath cotton at 9d. per lb. ready money; I demand at what price the cotton must be at in barter; also how much cotton must be bartered for 100 dozen of candles? Ans. The cotton at 9d. 3 grs. per lb. and 7 cwt. O 16 lb. of cotton must be given for 100 dozen of candles. 86. If a clerk's salary be £73 a year, what is that per day? Ans. 43. 37. B hath an estate of £53 per annum, and payeth 5s. 10d. to the subsidy, what must C pay whose estate is worth £100 per annum? Ans. 11s. Od.. 38. If I buy 100 yards of riband, at 3 yards for a shilling, and 100 more at 2 yards for a shilling, and sell it at the rate of 5 yards for 2 shillings, whether do I get or lose, and how much Ans. lose Ss. 4d 39. What number is that, from which if you take, the remainder will be? Ans. 29 40. A farmer is willing to make a mixture of rye at 4s. a bushel, barley at 3s. and oats at 2s. how much must he take of each to sell it at 2s. 6d. the bushel? Ans. 6 of rye, 6 of barley, and 24 of oats. 41. If of a ship be worth £3740, what is the worth of the whole? Ans. £9978...6...8. 42. Bought a cask of wine for £62...8, how many gallons were in the same, when a gallon was valued at 5s, 4d. Ans. 284. 43. A merry young fellow in a small time got the better of of his fortune; by advice of his friends he gave £2200 for an exempt's place in the guards; his profusion continued till he had no more than 880 guineas left, which he found by computation was part of his money after the commission was bought; pray what was his fortune at first? Ans. £10450. 44. Four men have a sum of money to be divided amongst them in such a manner, that the first shall have of it, the second, the third, and the fourth the remainder, which is 8, what is the sun? Ane, £112. |