41. A draper bought a piece of cloth at 3s. 2d. per yard. He sold one-third of it at 4s. per yard, one-fourth of it at 3s. 8d. per yard, and the remainder at 3s. 4d. per yard; and his gain on the whole was 14s. 2d. How many yards did the piece contain?

42. A grazier spent £33. 7s. 6d. in buying sheep of different sorts. For the first sort, which formed one-third of the whole, he paid 9s. 6d. each. For the second sort, which formed one-fourth of the whole, he paid 11s. each. For the rest he paid 12s. 6d. each. What number of sheep did he buy?

43. A market woman bought a certain number of eggs, at the rate of 5 for twopence; she sold half of them at 2 a penny, and half of them at 3 a penny, and gained 4d. by so doing: what was the number of eggs?

44. A pudding consists of 2 parts of flour, 3 parts of raisins, and 4 parts of suet; flour costs 3d. a lb., raisins, 6d., and suet 8d. Find the cost of the several ingredients of the pudding, when the whole cost is 2s. 4d.

45. Two persons, A and B, were employed together for 50 days, at 5s. per day each. During this time A, by spending 6d. per day less than B, saved twice as much as B, besides the expenses of two days over. How much did A spend per day?

46. Two persons, A and B, have the same income. A lays by one-fifth of his; but B by spending £60 per annum more than A, at the end of three years finds himself £100 in debt. What is the income of each?

47. A and B shoot by turns at a target. A puts 7 bullets out of 12 into the bull's eye, and B puts in 9 out of 12; between them they put in 32 bullets. How many shots did each fire?

48. Two casks, A and B, contain mixtures of wine and water; in A the quantity of wine is to the quantity of water as 4 to 3; in B the like proportion is that of 2 to 3. If A contain 84 gallons, what must B contain, so that when the two are put together, the new mixture may be half wine and half water?

49. The squire of a parish bequeaths a sum equal to one-hundredth part of his estate towards the restoration of the church; £200 less than this towards the endowment of the school; and £200 less than this latter sum towards the County Hospital. After deducting these lega39

cies, of the estate remain to the heir. What was the 40 value of the estate?

50. How many minutes does it want to 4 o'clock, if three-quarters of an hour ago it was twice as many minutes past two o'clock?

51. Two casks, A and B, are filled with two kinds of sherry, mixed in the cask A in the proportion of 2 to 7, and in the cask B in the proportion of 2 to 5: what quantity must be taken from each to form a mixture which shall consist of 2 gallons of the first kind and 6 of the second kind?

52. An officer can form the men of his regiment into a hollow square 12 deep. The number of men in the regiment is 1296. Find the number of men in the front of the hollow square.

53. A person buys a piece of land at £30 an acre, and by selling it in allotments finds the value increased threefold, so that he clears £150, and retains 25 acres for himself: how many acres were there?

54. The national debt of a country was increased by one-fourth in a time of war. During a long peace which followed £25000000 was paid off, and at the end of that time the rate of interest was reduced from 4 to 4 per cent. It was then found that the amount of annual interest was the same as before the war. What was the amount of the debt before the war?

55. A and B play at a game, agreeing that the loser shall always pay to the winner one shilling less than half the money the loser has; they commence with equal quantities of money, and after B has lost the first game and won the second, he has two shillings more than A: how much had each at the commencement?

56. A clock has two hands turning on the same centre; the swifter makes a revolution every twelve hours, and the slower every sixteen hours: in what time will the swifter gain just one complete revolution on the slower?

57. At what time between 3 o'clock and 4 o'clock is one hand of a watch exactly in the direction of the other hand produced?

58. The hands of a watch are at right angles to each other at 3 o'clock: when are they next at right angles?

59. A certain sum of money lent at simple interest amounted to £297. 12s. in eight months; and in seven more months it amounted to £306: what was the sum?

60. A watch gains as much as a clock loses; and 1799 hours by the clock are equivalent to 1801 hours by the watch: find how much the watch gains and the clock loses per hour.

61. It is between 11 and 12 o'clock, and it is observed that the number of minute spaces between the hands is two-thirds of what it was ten minutes previously: find the time.

62. A and B made a joint stock of £500 by which they gained £160, of which A had for his share £32 more than B: what did each contribute to the stock?

63. A distiller has 51 gallons of French brandy, which cost him 8 shillings a gallon; he wishes to buy some English brandy at 3 shillings a gallon to mix with the French, and sell the whole at 9 shillings a gallon. How many gallons of the English must he take, so that he may gain 30 per cent. on what he gave for the brandy of both kinds?

64. An officer can form his men into a hollow square 4 deep, and also into a hollow square 8 deep; the front in the latter formation contains 16 men fewer than in the former formation: find the number of men.

XXIII. Simultaneous equations of the first degree with two unknown quantities.

205. Suppose we have an equation containing two unknown quantities x and y, for example 3x-7y=8. For every value which we please to assign to one of the unknown quantities we can determine the corresponding value of the other; and thus we can find as many pairs of values as we please which satisfy the given equation. Thus, for example, if y=1 we find 3x=15, and therefore x=5; if y=2 we find 3x=22, and therefore x=7; and

so on.

Also, suppose that there is another equation of the same kind, as for example 2x+5y=44; then we can also find as many pairs of values as we please which satisfy this equation.

But suppose we ask for values of x and y which satisfy both equations; we shall find that there is only one value of x and one value of y. For multiply the first equation by 5; thus

Thus if both equations are to be satisfied x must equal 12. Put this value of x in either of the two given equations, for example in the second; thus we obtain

206. Two or more equations which are to be satisfied by the same values of the unknown quantities are called simultaneous equations. In the present Chapter we treat of simultaneous equations involving two unknown quantities, where each unknown quantity occurs only in the first degree, and the product of the unknown quantities does not occur.

207. There are three methods which are usually given for solving these equations. There is one principle common to all the methods; namely, from two given equations containing two unknown quantities a single equation is deduced containing only one of the unknown quantities. By this process we are said to eliminate the unknown quantity which does not appear in the single equation. The single equation containing only one unknown quantity can be solved by the method of Chapter XIX; and when the value of one of the unknown quantities has thus been determined, we can substitute this value in either of the given equations, and then determine the value of the other unknown quantity.

208. First method. Multiply the equations by such numbers as will make the coefficient of one of the unknown quantities the same in the resulting equations; then by addition or subtraction we can form an equation containing only the other unknown quantity.

This method we used in Art. 205; for another example, suppose

8x + 7y=100, 12x-5y=88.

If we wish to eliminate y we multiply the first equation by 5, which is the coefficient of y in the second equation, and we multiply the second equation by 7, which is the coefficient of y in the first equation. Thus we obtain