many as the men and women together. If a man cost more than a woman, and 3 children as much as a man and a woman together, and the whole cost for the month be £83 6s., how much is the daily cost of each man, woman, and child? 22. In 1858 the value of £100 tithe rent-charge, reckoned on the average price of corn in the 7 years immediately preceding, was £105; in 1859, reckoned in the like way, it is 3 per cent. more. If it were reckoned on the price in 1851 only, it would be but £69. What would it be if reckoned on the price in 1858 only? 23. Show that √2 lies between 12 and 11. 24. Extract the cube root of 669-921875 cubic feet, and reduce the result to inches. 25. What is the present worth of £1842 15s., payable a quarter of a year hence, at 5 per cent.? 26. What length of paper 22 inches wide would be required to paper the walls of a room 18 ft. 9 in. long, 13 ft. 3 in. broad, and 14 ft. 6 in. high? 27. Express 4 acres 2 roods 16 perches as the decimal of a square mile. 28. Multiply, by duodecimals, 7 ft. 3 in. 5 pts. by 5 ft. 7 in. 4 pts., and the product by 4 ft. 2 in. What does the product become when expressed in cubic feet and inches? 29. A man purchases £700 stock in the 3 per cent. Consols at 942, and also invests £585 in the purchase of Russian 5 per cent. stock at 97. How much stock has he standing in his name? If he sells out of the 3 per cents. at 95, and out of the 5 per cents. at 961, does he gain or lose by the transaction, and how much? 30. Divide 91-863 by 87·56. 31. A tradesman's stock in trade is valued on January 1st, 1868, at £8000, he has also £350 in cash and owes £1870; during the year his personal expenses, £300, are paid out of the proceeds of his business, and on January 1st, 1869, his stock is valued at £7950, he has £570 in cash and owes £1510. What is the whole profit on the year's transactions after deducting 5 per cent. interest on the capital with which he began the year? 32. A grocer buys coffee at the rate of £8 10s. per cwt. and chicory at £2 10s. per cwt., and mixes them in the proportion of 5 parts of chicory to 7 of coffee. At what rate must he sell the mixture so as to gain £163 per cent. on his outlay? 33. Two clocks point to 2 o'clock at the same instant on the afternoon of Christmas-day; one loses 8 seconds, and the other gains 9 seconds, in 24 hours. When will one be half an hour before the other, and what time will each clock then show? 34. Three sacks contain equal quantities of wheat, barley, and oats, and a mixture is formed by taking 60 per cent. from the 1st sack, 40 per cent. from the 2nd, and 70 per cent. from the 3rd. Find what percentage of the whole quantity of corn is taken away. A candidate being required to answer this question reasons as follows: '50 is the mean between 60 and 40, and 60 is the mean between 50 and 70, therefore 60 is the mean percentage required.' Is this reasoning erroneous? If so, show clearly where the error lies. 35. If the light received by an object at different distances from a candle be inversely as the squares of the distances, and a mirror reflect of the light which falls on it, at what distance from the candle must an object be placed, in order to receive the same amount of light directly, which it would receive by reflection from the mirror so placed that the whole distance traversed by the light is 5 feet? of 36. A contractor has stores worth £300; he sells them so as to lose 10 per cent. What must he sell the remainder for so as to gain 20 per cent. on the whole? 37. (1) How many cubic feet of lead inch thick will be required to cover the sides and bottom of a cistern 10 ft. long, 6 ft. 6 in. wide, and 7 ft. deep? (2) What weight of water will the cistern hold, a cubic foot of water weighing 1000 ounces? 38. If 2 cubic inches of iron weigh as much as 15 of water, and a cubic foot of water weighs 1000 oz., find the weight of a beam of iron 5 yds. long, 2 ft. wide, and 1 in. thick. 39. Two workmen contract to do a piece of work for £25, which they reckon will take them 75 days to execute. After working together 45 days, one of them falls ill, and his mate completes the job in 50 days more. How much is each entitled to receive? 40. A contractor engaged to finish a work of 3150 cubic yds, in 50 days, and employed at once 60 men upon it, but at the end of 35 days he finds only 1800 cubic yds. completed. How many extra men must he put on to complete the work in the given time? 41. If a person purchase £4000 stock at 93%, what loss will he sustain by selling out at 927? 42. How many square feet of board will be required to make a cubical box which shall contain 81.370 cubic feet? 43. If a new unit of weight were instituted containing 1 lb. 1 oz. 3·326 drs. avoirdupois, how many of such units of weight would make 537 lbs. 11 oz. 15 drs.? 44. If a company, when its capital was £2000000, paid 7 per cent. to the shareholders, what is its capital now when it can pay no more than 3 per cent., although the profit to be divided is six times as much as it was in the former case? 45. If gold be beaten out so thin that 1 oz. will form a leaf of 20 square yards, how many of these leaves will make '01 in. thick, if 1 cubic foot of gold weighs 1095 lbs.? 46. The stuff out of a lead mine contains at first 15.9 per cent. of lead; after washing, by which process the amount of lead ore is not diminished, the stuff contains 87-45 per cent. of lead; how much rock was washed away out of 216 tons 5 cwts. of the original stuff? 47. A field is 300 yds. long, 200 yds. broad; if a belt of trees 30 yds. wide be planted round it, find the area of the Find also the distance from corner to interior space. corner. 48. A fraudulent tradesman sells a piece of cloth for 31 yds., at 19s. 6d. a yard; what is his profit from having used a yard measure 1 in. too short? 49. An artisan going to his work every week-day is obliged to pay d. toll, the toll being converted into one of 2 mils. How much will he gain in a year by the change of currency? 50. A sovereign, standard gold, weighs 5·136 dwts.; a shilling, standard silver, weighs lb. troy. What value to 4 oz. of weight of standard silver is equal in standard gold? 51. If 9 tons 7 cwts. of iron be sold for £245, and the gain upon it be 20 per cent., what was the cost of 1 cwt.? ALGEBRA. MULTIPLICATION AND DIVISION. 1. When m and n (m>n) are whole numbers, show that am xan=am+n ; am÷an=am−n ; ao=1 ; and that 2. What is the meaning of the term factor? x-1 is a factor of x2+x-2, what is the other factor? 3. Divide (1) x5—3x1+4x3 +26x2−92x+55 by x2−3x+11. (4) x—4y by x3—2y3. (5) (x3-12x+16) × (x3-12x-16) by x2-16. 4. If xz+yz—xy=2c2, xz+xy-yz=2a2, xy+yz—xz=2b2, show that x2=(a2+c2) (a2+b2) 5. Resolve into factors b2+c2 25x2-16; x3-8; x3+27y3; a3x3-64b3. 6. Divide (1) hkx1+2 (h−k) x3 — (h2+4—k2) x2+2 (h+k)x-hk by kx2-h+2x. (2) mn (x2+1)+(m2+n2) (x2+x)+(n2+2mn) (x3+x2) by nx2+mx+n. (3) (b−c) a3+(c− a) b3 + (a−b) c3 by (a−c)(a−b). aab 3 7. Multiply + + by 3 + 4 2 862 4 2 862 8. If a3—pa2+qa—r=0, then x3 —px2+qx—r is divisible by x-a. 9. Divide (1) (x2+xy+y2)× (x3+y3) by x+x2y2+y^. (3) a2-b2+c2-d2-2 (ac-bd) by a-c+b-d. 10. (1) Prove that— {(a-b)2+(b-c)2+(c-a)2} 2 |