which they received was to that which the others received, as 6 to 5. Required the value of a share in each company. 35. A alone could perform a piece of work in 12 hours; A and C together could do it in 5 hours; and C's work is of B's. Now, the work has to be completed by mid-day. A begins work at 5 o'clock in the morning; at what hour ought B and C to relieve him so as to finish the work in time? 36. At what hour ought they to join him, so that the three working together may complete it in time? 37. A person dying left his property equally between his two sons. After seven years, the one had quadrupled his money, and the other had lost £1000, and it was found that the former possessed five times as much as the latter. Required the sum left. 38. It is required to divide 77 into four such parts that if the first be diminished by 2, the second increased by 2, the double of the third increased by 1, and three-fifths of the fourth diminished by 4, the results shall all be equal. 39. When a body of men are to be formed into a square, it is found that there are 21 men over, but when it is attempted to add one man to each side of the square, it is found there are 200 men too few. Required the number of men. 40. Divide £1000 between A, B, C, and D, giving to A twice as much as to B, to B three times as much as to C, and to C four times as much as to D. 41. The assets of a bankrupt's estate are found to pay a dividend of 7s. 9d. in the pound, on the ascertained debts, leaving a balance of £99: 15s. But a new claim having been made, it is found that to pay a dividend of 7s. 9d. in the pound, the assets are deficient by £16:10s. Required the amount of the new claim. 42. The rise of interest from 3 to 4 per cent increases a person's net income (after deducting income tax of 7d. in the pound) by £485: 8:4. What is the principal sum from which his income is derived? 43. The assets of a bankrupt's estate are just sufficient to pay a dividend of 16s. 7d. in the pound, but the expense of realizing amounts to £1100, and it is found that after deducting this, the dividend is reduced to 14s. 7d. in the pound. Required the debt. 44. In 1852 income-tax was 7d. in the pound; in 1860 it is 10d. A man's clear income in 1852, after paying the tax, was £466, what will it be in 1860 ? 45. Of every 24 oz. of what is called gold, only 18 in jewellery, and 22 in sovereigns are really gold. A sovereign weighs 123 grains. When a jeweller offers to sell gold chains for their weight in sovereigns, what does he charge per oz. for workmanship, etc. 46. A man travels at the rate of 4 miles an hour when his day's journey is to last only 7 hours; but if he is required to continue his journey for a longer period, he decreases his speed by a quarter of a mile per hour throughout the day for every additional hour he is required to travel. Having a long journey to perform, he divides his labour into three periods, each of the same number of days-during the first period he travels 9 hours a day; during the second, 8; and during the third 7. Now he finds that had he performed the whole journey at the same rate of 8 hours a day, he would have had two hours to spare on the last day. Required the length of his journey. 47. Gold is 19 times as heavy as water, and silver 101 times. A mixed mass weighs 4160 oz., and displaces 250 ounces of water. What proportions of gold and silver does it contain ? The 250 ounces are its bulk. Let x be the bulk of gold, 250x of silver. 48. A rectangular court, in the form of a double square, has a pavement 3 feet wide running quite round it. The area of the pavement is 828 square feet. Required the dimensions of the court. Ex. 22. 1. Three hats and five caps cost £5: 4s.; four hats and seven caps cost £7: 2s. Required the cost of a hat and a cap respec tively. 2. The wages of five men and four women amount to £12:16s.; and seven men receive more than six women by £4. What does each receive? 3. A workman having worked 12 days, and been idle 5, received 25s. When he worked 16 days, and was idle 7, he got 33s. What was his daily wage, and the charge for his maintenance ? 4. What fraction is that which, when its numerator is increased by 7, becomes equal to §, and when its denominator is increased by 10, equal to 5. A person distributing money among some poor people, found he wanted 10s. to be able to give 5s. to each; he therefore gave them each 4s., and found he had 5s. left. the number of shillings, and the number of people. Required He 6. A person takes 15 horses and 100 sheep to market. sells 8 of the horses and all the sheep to a dealer for £326. But, in the after part of the day, finding himself unable to dispose of the remaining horses, he arranges with the same dealer to take them, on condition of his receiving back half the sheep at a loss of 2s. a head. He finds that his total receipts on the transactions amount to £411. Required the price of a horse and of a sheep. 7. A person takes 12 horses and a flock of sheep to market. He first sells 8 horses and of his sheep, the latter at £1 : 4s. a head, for £262. He then disposes of the remainder of his stock, the horses at a reduction of £1 a head, the sheep at a reduction of 4s. a head, for £157. Required the price of a horse, and the number of sheep. 8. There is a number composed of two digits, whose sum is 7. If we add 27 to the number, the digits will be reversed. Required the number. 9. A, B, and C together possess £60; A, B, and D together £72; A, C, and D together £90; and B, C, and D together £102. Required what each possesses. Add the four equations together, and divide by 3. From the result subtract each of the equations separately. 10. If A and B each gain £24, A will be twice as rich as B; but if each lose £24, A will be three times as rich as B. Required what each has. 11. A and B have between them £100; C and D between them £120. A is twice as rich as C, and D three times as rich as B. What does each possess? 12. A wine-merchant sold 3 dozen of brandy and 5 dozen of wine for £22: 16s. Had the sale been 5 dozen of brandy and 3 dozen of wine, the bill would have been £25: 4. Required the price of brandy and of wine. 13. A boy at a fair spent his money in oranges. If he had got five more for his money, they would have averaged a halfpenny each less; and if three less, a halfpenny each more. How much did he spend ? 14. A loses of his money, and B gains as much as of A's money. They then find themselves equally rich. But on A losing £50 more, and B gaining as much, A is only half as rich as B. Required what each had. 15. The dimensions of a rectangular court are such, that if its length were increased by 1 yards, and its breadth diminished by the same, its area would be diminished by 231 yards; and if its length were increased by 1 yards, and its breadth increased by the same, its area would be increased by 524 yards. Required its dimensions. 16. There are three metallic compositions. The first is such, that 16 oz. of it contain 7 of gold, 8 of silver, and 1 of copper; the second such, that 16 oz. contain 5 of gold, 7 of silver, and 4 of copper; the third such, that 16 oz. contain 2 of gold, 9 of silver, and 5 of copper. Required out of these to compose a fourth such, that 16 oz. shall contain 411⁄2 oz. of gold, 718 of silver, and 3 of copper. 17. Income and assessed taxes together amount to £30. But if income tax be increased 20 per cent, and assessed taxes diminished 25 per cent, the taxes will together amount to £32 26. Required each. : 18. A cistern which holds 130 gallons is supplied by two pipes. After the first has been open for 20 minutes, the other is opened, and the two fill the cistern in 25 minutes more. If the second had been set agoing for 20 minutes instead of the first, the two would have completed the filling in 20 minutes more instead of 25. Required the number of gallons which each supplies per minute. 19. A cistern which contains 150 gallons, is supplied by one pipe and emptied by another. If the first is turned on for 20 minutes, and then the second, the whole time occupied in filling the cistern will be 2 hours and 5 minutes. If the second pipe had been turned on half an hour later, the time saved would have equalled half that during which, in the last case, the two are open together. Required the number of gallons which flow through each per minute. 20. A slow train starts from Edinburgh to London (410 miles), 3 hours before the express. The latter overtakes it at a distance of 129 miles from Edinburgh. Two hours after the arrival of the express in London it returns on its journey, and meets the slow train 1043 miles from London. Required the rates of travelling of the two trains. 21. A, B, C and D play at cards, beginning each with the same sum. First A and B win each the same sum from C and D. Next A and C lose each half that sum to B and D. Thirdly, A and D win each a fourth that sum from B and C. Lastly, A wins 2s. from D. It is then found that A and B have each as much as C and D together. Required the sum with which each commenced play, and the sum first lost. |