47. There are three numbers, the difference of whose differences is 5; their sum is 44, and continued product 1950. Find the numbers. Ans, 25, 13, and 6.

48. Having sold goods for £24, I gained as much per cent as the whole cost me. Required the cost.

49. There are two square buildings, that are paved with stones a foot square each: the side of one building exceeds that of the other by 12 feet; and both their pavements taken together, contain 2120 stones. lengths separately?

What are their

Ans. 26 and 38.

£72, and found,

50. A person bought some sheep for that if he had bought 6 more for the same money, he would have paid £1 less for each. Required the number bought, and the price of each.

Ans. The number was 18, and the price of each £4. 51. The difference between the hypotenuse and base of a right-angled triangle is 6; and the difference between the hypotenuse and perpendicular is 3. Required the

sides.

Ans. 9, 12, 15.

52. There is a field in the form of a rectangle, whose length is to its breadth as 6 to 5; one-sixth of the whole area being planted, there remain for ploughing 625 square yards. Required the dimensions of the field.

Ans. 30 by 25.

53. Two messengers were dispatched at the same time to a place 90 miles distant; the former of whom, by riding one mile an hour faster than the other, arrived at the end of his journey an hour before the other. Required the rate of each per hour.

Ans. 10 and 9 miles per hour. 54. A regiment of soldiers, consisting of 1066 men, is formed into two squares, one of which has 4 men more

in a side than the other. Required the number in each side. Ans. 21 and 25.

55. A company at a tavern had £8 15s. to pay; but before the bill was paid, two of them went away, and in consequence, those who remained had each 10s. more to pay. How many persons were in the company at first? Ans. 7 persons.

56. Find two numbers whose product is 300, and such that if 10 be added to the less, and 8 subtracted from the greater, the product of the sum and remainder shall also be 300. Ans. 20 and 15.

57. A labourer is engaged for n days, on condition that he receives p pence for every day he works, and pays q pence for every day he is idle; at the end of the time he receives a pence. How many days did he work, and how many was he idle?

58. A poulterer bought 15 ducks and 12 turkeys for 5 guineas, and had two ducks more for 18s. than he had of turkeys for 20 shillings. What was the price of each? Ans. 3s. and 5s.

59. The sum of the numerator and denominator of a fraction is equal to 6 times the fraction; and the sum of the numerator and denominator of its reciprocal is equal to 1 times its reciprocal. What is the fraction?

Ans..

60. It is required to find the three sides of a right-angled triangle, from the following data: the number of square feet in the area is equal to the number of feet in the hypotenuse, together with the sum of the feet in the other two sides; and the square described upon the hypotenuse

s less than the square described upon the sum of the two sides, by half the product of the base and area.

Ans. Hypotenuse 10, the sides 6 and 8. 61. A man travelled 105 miles, and found that, if he had not travelled so fast by 2 miles an hour, he would have been 6 hours longer in performing the same journey. How many miles did he go per hour?

Ans. 7.

62. A farmer bought two flocks of sheep for £65 13s. one containing 5 more than the other: each sheep cost as many shillings as there were sheep in the flock. Required the number in each. Ans. 23 and 28.

63. The joint stock of two partners, A and B, was £416; A's money was in trade 9 months, and B's 6 months when they shared stock and gain, A received £228, and B £252. Required each man's stock.

Ans. The stocks are £192 and £224.

64. Two persons, A and B, travelled on the same road and at the same rate to London. At the 50th milestone from London A overtook a flock of geese, which travelled at the rate of 3 miles in 2 hours, and 2 hours afterwards he met a stage waggon which travelled at the rate of 9 miles in 4 hours. B overtook the flock of geese at the 45th milestone from London, and met the stage waggon 40 minutes before he came to the 31st milestone. Where was B when A reached London ?

Ans. 25 miles from London. 65. In comparing the rates of a watch and a clock, it was observed, on one morning, when it was 12 by the clock that the watch was at 59 minutes 49 seconds past 11; and, two mornings after, when it was 9 by the clock, the watch was at 59 minutes 58 seconds past 8. The clock is known to gain second in 24 hours. Find the gaining rate of Ans. 4.9 seconds in 24 hours.

the watch.

66. If A and B together can perform a piece of work in a days, A and C together in b days, and B and C together in c days, find the time in which each can perform the work separately.

67. A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished 4 times as many men as there were companies in the regiment; but these being found to be insufficient, each company furnished 3 more men, when their number was found to be increased in the ratio of 17 to 16. How many companies were in the regiment?

Ans. 12 companies.

68. The product of the sum and difference of the hypotenuse and a side of a right-angled triangle is equal to 2; and 4 times the sum of the squares of the hypotenuse and this side is equal to 5 times the sum of these two lines. Find the three sides of the triangle.

Ans. 1,, and 5

236

CHAP. X.-METHOD OF INDETERMINATE
COEFFICIENTS.

ART. 149. If it were required to resolve the fraction a into an infinite series, without actually dividing,

a+bx

we at once see (Art. 81) that the series will consist of the powers of a multiplied by certain coefficients, the indices of the powers of x, beginning at zero and increasing by a unit in each successive term. That such development is possible, will also appear from Art. 92, since the given expression is the same as a(a+bx)-1. It is not, however, obvious what the coefficients will be; but it is plain that they are independent of x, and are expressed in terms of a and b. Let us then assume the coefficients to be A, B, C, D, &c. and we shall

have

a

a+bx

=1+Ax+Bx2+Сx3+Dx1, &c.,

an identical equation, the assumed series being merely the supposed result of the operation indicated by the left member. To determine these coefficients, multiply both members by the denominator, and arrange the terms by the powers of x; we thus obtain

a=a+Aa | x+Ba | x2+Ca | x3+Da | x4 &c. *

* That the left member is of the same form as the right, will be plain when it is written thus, a+0x+0x2+0x3, &c.