MISCELLANEOUS QUESTIONS.* 1. Ir a person gain 8 per cent. by selling apples at the rate of 8 for 6 d., how much does he gain per cent. by selling them at the rate of 3 for 24d? Answer, 11}. 2. If eggs be bought at the rate of 5 for a penny, how must they be sold to gain 40 per cent.? Answ. At the rate of 25 for 7d. 3. If 150 apples cost 9/4, how many of them must be sold at the rate of 8 for 6 d., and how many at the rate of 3 for 24d., that the gain on the whole may be 10 per cent.? Answ. 90 at 3 for 24d., and 60 at 8 for 64d. 4. A merchant engages a clerk at the rate of £20 for the first year, £25 for the second, £30 for the third, &c., thus augmenting his salary by £5 each year. How long must the clerk retain his situation, so as to receive on the whole as much as he would have received, had his salary been fixed at £52 10 annum? Answ. 14 years. 5. Three gentlemen contribute £164 5 towards the building of a church at the distance of 2 miles from the first, 23 miles from the second, and 33 miles from the third; and they agree that their shares shall be reciprocally proportional to their distances from the church. How much must they severally contribute? Answ £72 9, £50 8, and £41 8. 6. If a person purchase pins, when there are 18 in the row, which sells for a farthing, and sell them when there are only 11 in the row, how much is his gain cent.? Answ. 6311. 7. A hosier sells 90 pair of stockings and gloves for £12 10, the stockings at 3/, and the gloves at 2/6 pair. Required the number of each. Answ. 50 pair of stockings, and 40 pair of gloves. 8. A son having asked his father's age, the father replied: "Your age is twelve years; to which if five-eighths of both our ages be added, the sum will be equal to mine." What was the father's age? Answ. 52 years. * The questions contained in this article are intended to exercise the advanced stu. dent in the use of the several rules and modes of operation, exhibited both in the text and in the notes of the preceding part of this work. They are not adapted for the ajority of arithmetical pupils; as for their purpose they are too difficult, and possess ton little practical utility. It is hoped, however, that, besides affording inuch practice in calculation, and in the application of the rules already delivered, they will form useful exercises for the reasoning powers of those who have taste or ability for such speculations. The great and principal object with every teacher of Arithmetic, should be, to make his pupils acquire an extensive and substantial practical knowledge of this science, without occupying their time and attention with many puzzling or difficult questions. At the same time, however, when he meets with pupils of capacity, and of considerable proficiency, it may be very proper to direct their attention to such questions as are contained in this article. By this means he will have a farther proof of their capacity, and he may lay the foundation of future proficiency in other departments of mathematical science. Some of the following questions may be solved perhaps most casily by Position. They may all, however, be wrought without Position: and the student should endeavour to do them without this ruie, as they will thus, form a much better exercise for his thinking powers, and fit him in a greater degree, for solving questions of difficulty, either in Arithmetic, or in other departments of Mathematics. 9. The population of Great Britain was 10,820,100 in 1801, and 12,596,803 in 1811. Hence, it is required to find its amounts in 1810 and 1821, the yearly increase being supposed to be proportional to the population. Answ. 12,406,733 in 1810, and 14,665,248 in 1821. 10. Three merchants having formed a joint stock of £1064, A's stock continues in trade months, B's 8 months, C's 12 months; and A's share of the gh is £114, B's £133 4, and C's £165. What was the stock of each? Answ. A's £456, B's £333, and C's £275. 11. The stocks of three partners, X, Y, and Z, continue in trade 8, 10, and 7 months respectively; and their respective gains are £115 10, £204 15, and £183 15. Hence, it is required to find their several stocks, the difference between those of Y and Z being £220. Answ. X's stock £550, Y's £780, and Z's £1000. 12. The joint sum of two series' of continual proportionals, con sisting of five terms each, and having a common mean, is 8037, and their ratios are 1 and 2. Required the series'. Answ. 24, 34, 5, 73, 111; and 4, 2, 5, 125, and 314. 13. If A, B, and C could pave a street in 18 days; B, C, and D in 20 days; C, D, and A in 24 days; and D, A, and B in 27 days; in what times would it be done by all of them together, and by each of them singly? Answ. By all in 16 days; by A in 87 days; by B in 50% days; by C in 417 days; and by D in 1701 days. 14. If A can reap a field in 13 days, and B in 16 days, in what time would both together reap it? Answ. In 7 days. 15. If A and B, with C working half time, could build a wall in 21 days; B and C, with D working half time, in 24 days; C and D, with A working half time, in 28 days; and D and A, with B working half time, in 32 days; in what times would it be built by all of them together, and by each of them singly? Answ. A would finish it in 52 days; B in 5733 days; C in 44 days; D in 280 days; and all in 16 days. 16. X, Y, and Z can build of a wall in 10 days; Y, Z, and V, of it, in 6 days; Z, V, and X, of it in 7 days; and V, X, and Y, the remainder of it in 9 days. Hence, it is required to find in what times it would by done by all of them together, and by each of them separately. Answ. By all in 23,5 days; and by X, Y, Z, and V respectively, in 2921, 7984388, 9333333, and 6248183 days. 7849 17. The stocks of three partners, A, B, and C, are £350, £220, and £250, and their gains £112, £88, and £120, respectively; and B's stock continued in trade 2 months longer than A's. Required the time the money of each continued in trade. Answ. 8, 10, and 12 months respectively. 18. In what arithmetical scale would five hundred and fifty four be expressed by 95? Answ. In the scale whose radix is 61. 19. The population of Great Britain was 6,523,000 in 1700; 7,860,000 in 1750; 10,820,100 in 1801; and 12,596,803 in 1811. Hence it is required to find the annual rates of increase on each million of the population between the first and second, the second and third, and the third and fourth, of the above-mentioned years, the rate of increase at any time during each period, being supposed to be proportional to the population at that time. Answ. During the first period, 3736; during the second, 6287; and during the third, 15320. 20. If a gallon of water were resolved into the oxygen and hy. drogen of which it is composed, it is required to determine the bulk into which it would thus be expanded, water being 741 times heavier than an equal bulk of oxygen, and 9699 times heavier than an equal bulk of hydrogen. (See exer. 5, p. 192.) Answ. The two gases would fill 17944 gallons. 21. The Neperian logarithm* of any number, is to the common logarithm of the same number as 1 is to 43429448, nearly. Required the series of ratios converging to this ratio. Answ. 2 to 1; 7 to 3; 23 to 10; 76 to 33; 99 to 43; 175,to 76; (or 700 to 304;) 624 to 271; 3919 to 1702; 12381 to 5377; 16300 to 7079, &c. 22. Reduce the fraction whose numerator is the square root of 5, and denominator the square root of 11, to a continued fraction, and find the first seven of the series of fractions converging to its value. Answ. 1, 3, 43, 88, 111, 331, and $191. 358 23. A person, in discounting a bill, at 6 per cent. per annum, according to the common or false method, finds that he has 6 per cent. per annum for his money. How long must the bill have been discounted before it was due? Answ. 1 year and 103 days, nearly. 24. A person pays £54 for the insurance of goods at 33 per cent.; and he finds, that in case of the goods being lost, he will by this means be entitled to the value of the goods, the premium of insurance, and £5 besides. What is the value of the goods? Answ. £1381. 25. Reduce five ninths to a fraction in the septenary scale of notation, whose denominator in that scale may be expressed by a unit with as many ciphers annexed as there are figures in the numerator. Answ. The numerator will be 361361361, &c. or 3'61'. 26. If a merchant each year increase his capital by a fifth part of itself, except an expenditure of £400 per annum, and at the end of 15 years be worth £12000; find, without Position, his original capital. Answ. £2649 1 1. 27. A servant draws off one gallon each day for 20 days, from a cask containing 10 gallons of rum, each time supplying the deficiency by the addition of a gallon of water; and then, to escape * Neperian logarithms are also frequently, but improperly, called hyperbolic logarithms. The number 4342944319, &c. is called the modulus of the system of common logarithins, detection, he again draws off 20 gallons, supplying the deficiency each time by a gallon of rum. It is required to determine how much water still remains in the cask. Answ. 1.0679577 gallon, or rather more than a gallon and half a pint. 28. A sells a quantity of tea, which cost him £246 12, to B; and B sells it to C, who disposes of it for £391 11 10. Required the prices at which A and B sold it, each of the three merchants having gained at the same rate per cent. Answ. A sold it for £287 14, and B for £335 13. 29. A and B set out from the same place, and in the same diection. A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days; he then turns again, and pursuing his journey, overtakes B 22 days from the time they first set out. It is required to find the rate at which B uniformly travelled. Answ. 10 miles per day. 30. A merchant every year gains 50 per cent. on his capital, of which he spends £300 per annum in house and other expenses, and at the end of 4 years he finds himself possessed of a capital 4 times as great as what he had at commencing business. Find his original capital without using the rule of Position. Answ. £2294,7. 31. At what time does the sun set, when the length of the day (from sunrise till sunset) is four times the length of the morning or evening twilight, and the evening twilight two sevenths of the time from its termination till day-break? Answ. At 3 minutes past 5 o'clock. 32. How will 13579 in the trigesimal scale, be expressed in the duodecimal scale? Answ. 372433. 33. Find the first nine fractions approaching to the ratio of 1 to the cube root of 2. Answ. 1, 1, 1, 23, 37, 88, £87, 173, and 504 34. Required, the approximate ratios of the English foot to the French metre, and also to the toise. (See note, page 43.) Answ. The foot to the metre, as 1 to 3, 3 to 10, 4 to 13, 7 to 23, 25 to 82, 32 to 105, 57 to 187, 89 to 292, 146 to 479, &c.; and the foot to the toise, as 1 to 6, 2 to 13, 3 to 19, 5 to 32, 33 to 211, &c. 35. How will that fraction be expressed in the decimal scale, whose denominator in the octary scale is a unit with as many ciphers annexed as there are figures in its numerator, and its numerator 644 repeated without end? Answ.. 36. What is the difference between in the quinary scale, and in the nonary scale? Answ. Twenty-seven one-hundred-and thirty-thirds. 37. What is the product of in the duodecimal scale, and in the octary scale? Answ. One and five-elevenths. 38. It is required to find a sum of money, of which, in the space 4 years, the true discount, at simple interest, is £5 more at the rate of 6 than of 4 per cent. per annum. Answ. £89 18. 39. One third of a quantity of flour being sold to gain a certain rate per cent., one-fourth to gain twice as much per cent., and the remainder to gain three times as much per cent.: it is required to determine the gain per cent. on each part, the gain upon the whole being 20 per cent. Answ. The gains per cent. are 9§, 19, and 284. 40. A man travels from his own house to Belfast in 4 days, and home again in 5 days, travelling each day, during the whole journey, one mile less than he did the preceding. How far does he live from Belfast? Answ. 90 miles. 41. What is the radix of the arithmetical scale, in which 9(20) (12)609 in the trigesimal notation will be expressed by 5000004? Answ. 19. 42. The men employed by a gentleman work 12 hours, the women 9 hours, and the boys 8 hours, each day: for labouring the same number of hours, each man receives a half more than each woman, and each woman a third more than each boy: the entire sum paid to all the women each day is double of the sum paid to all the boys; and for every five shillings earned by all the women each day, twelve shillings are earned by all the men. Hence it is required to find the number of each class employed, the entire number being 59. Answ. 24 men, 20 women, and 15 boys. 43. A man leaves to his eldest child one-fourth of his property; to his second, one-fourth of the remainder, and £350 besides; to his third one-fourth of the remainder and £975; to his youngest one-fourth of the remainder and £1400; and what still remains he bequeaths to his wife, whose share is found to be one-fifth of the whole. Hence it is required to find the value of the whole property. Answ. £20,000. 44. The less of two bales of cloth is bought at the rate of twice as many pence per yard as it contains yards, and costs 31 0 2 more than the greater, which contains 4 yards for every 3 in the less, and is bought at the rate of as many pence per yard as it contains yards. How many yards are contained in each ? Answ. 244 yards in the greater, and 183 in the less. 45. It is required to find a sum of money such that its true discount for one year at 5 per cent. will be £1 more than the sum of the true discounts of one-half of it at 4 per cent. and the rest at 6 per cent. Answ. £11575 4. 46. A property of £10,000 is left to four children whose ages are 6, 8, 10, and 12 years respectively; and it is divided among them in such a manner, that their several shares being improved at 4 per cent. per annum, compound interest, they shall all have equal properties at the age of 21. It is required to determine the sum left to each. Answ. £2180 3 4†, £2380 15 114, £2599 17 93, and £2839 2 102. 47. A property is left to four children, one aged 6 years, two aged 9 years each, and one aged 11 years, in such a manner that all their properties are to be equal on their coming to age, compound interest being allowed at 4 per cent. per annum. Now, one |