toward A. The couriers met 7 hours after the second courier started. If the second courier had started from B 2 hours before the first started from A, they would have met 8 hours after the second courier started. At what rate did each courier ride?

43. A farmer has enough feed for his oxen to last a certain number of days. If he were to sell 75 oxen, his feed would last 20 days longer. If, however, he were to buy 100 oxen, his feed would last 15 days less. How many oxen has he, and for how many days has he enough feed?

44. An alloy of tin and lead, weighing 40 pounds, loses 4 pounds in weight when immersed in water. Find the amount of tin and lead in the alloy, if 10 pounds of tin lose 18 pounds when immersed in water, and 5 pounds of lead lose .375 of a pound.

45. Two men were to receive $96 for a certain piece of work, which they could do together in 30 days. After half of the work was done, one of them stopped for 8 days, and then the other stopped for 4 days. They finally completed the work in 35 days. How many dollars should each one receive, and in what time could each one have done the work alone?

46. Two boys, A and B, run a race from P to Q and return. A, the faster runner, on his return meets B 90 feet from Q, and reaches P 3 minutes ahead of B. If he had run again to Q, he would have met B at a distance from Pequal to one-sixth of the distance from P to Q. How far is from P, and how long did it take B to run from P to Q and return?

47. It took a certain number of workmen 6 hours to carry a pile of stones from one place to another. Had there been 2 more workmen, and had each one carried 4 pounds more at each trip, it would have taken them 1 hour less to complete the work. Had there been 3 fewer workmen, and had each one carried 5 pounds less at each trip, it would have taken them 2 hours longer to complete the work. How many workmen were there, and how many pounds did each one carry at every trip?

48. Three carriages travel from A to B. The second carriage travels every 4 hours 1 mile less than the first, and is 4 hours longer in making the journey. The third carriage travels every 3 hours 14 miles more than the second, and is 7 hours less in making the journey. How far is B from A, and how many hours does it take each carriage to make the journey? 49. Water enters a basin through one pipe and is discharged through another. Through the first pipe four more gallons enter the basin every minute than is discharged through the second. When the basin is empty, both pipes are opened, the first one hour earlier than the second, and after a certain time the basin contains 1760 gallons. The pipe through which water enters is then closed, and after one hour is again opened. If

both pipes be then left open as long as they were open together in the former case, the basin will contain 880 gallons. In what time can the one pipe fill the basin and the other empty it, if it hold 1760 gallons?

50. A body moves with a uniform velocity from a point A to a point B, which is 323 feet distant from A, and without stopping returns at the same rate from B to A. A second body leaves B 13 seconds after the first leaves A, and moves toward A with a uniform but less velocity than the velocity of the first. The first body meets the second 10 seconds after the latter starts, and on returning to A overtakes the second body 45 seconds after the latter starts. What is the velocity of each body?

51. A fox pursued by a dog is 60 of her own leaps ahead of the dog. The fox makes 9 leaps while the dog makes 6, but the dog goes as far in 3 leaps as the fox goes in 7. How many leaps does each make before the dog catches the fox?

52. The sum of the three digits of a number is 14; the sum of the first and the third digit is equal to the second; and if the digits in the units' and in the tens' place be interchanged, the resulting number will be less than the original number by 18. What is the number?

53. The sum of the ages of A, B, and C is 69 years. Two years ago B's age was equal to one-half of the sum of the ages of A and C, and 10 years hence the sum of the ages of B and C will exceed A's age by 31 years. What are the present ages of A, B, and C ?

54. The total capacity of three casks is 1440 quarts. Two of them are full and one is empty. To fill the empty cask it takes all the contents of the first and one-fifth of the contents of the second, or the contents of the second and one-third of the contents of the first. What is the capacity of each cask?

55. Three brothers wished to buy a house worth $70,000, but none of them had enough money. If the oldest brother had given the second brother one-third of his money, or the youngest brother one-fourth of his money, each of the latter would then have had enough money to buy the house. But the oldest brother borrowed one-half of the money of the youngest and bought the house. How much money had each brother?

56. The sum of the three digits of a number is 9. The digit in the hundreds' place is equal to one-eighth of the number composed of the two other digits, and the digit in the units' place is equal to one-eighth of the number composed of the two other digits. What is the number?

57. Find the contents of three vessels from the following data: If the first be filled with water and the second be filled from it, the first will then contain two-thirds of its original contents; if from the first, when full, the third be filled, the first will then contain five-ninths of its origi

nal contents; finally, if from the first, when full, the second and third be filled, the first will then contain 8 gallons.

58. Three boys were playing marbles.

A said to B: "Give me

5 marbles, and I shall have twice as many as you will have left." B said to C: "Give me 13 marbles, and I shall have three times as many as you will have left." C said to A: "Give me 3 marbles, and I shall have six times as many as you will have left." How many marbles did each boy have?

59. Three cities, A, B, and C, are situated at the vertices of a triangle. The distance from A to C by way of B is 82 miles, from B to A by way of C is 97 miles, and from C to B by way of A is 89 miles. How far are A, B, and C from one another?

60. A father's age is twenty-one times the difference between the ages of his two sons. Six years ago his age was six times the sum of his sons' ages, and two years hence it will be twice the sum of their ages. Find the ages of the father and his two sons.

61. A regiment of 600 soldiers is quartered in a four-story building. On the first floor are twice as many men as are on the fourth; on the second and third are as many men as are on the first and fourth; and to every 7 men on the second there are 5 on the third. How many men are quartered on each floor?

62. The sum of the three digits of a number is 9. If 198 be added to the number, the digits of the resulting number are those of the given number written in reverse order. Two-thirds of the digit in the tens' place is equal to the difference between the digits in the units' and in the hundreds' place. What is the number?

63. Four men are to do a piece of work. A and B can do the work in 10 days, A and C in 12 days, A and D in 20 days, and B, C, and D in 7 days. In how many days can each man do the work, and in how many days can they all together do the work?

64. The year in which printing was invented is expressed by a number of four digits, whose sum is 14. The tens' digit is one-half of the units' digit, and the hundreds' digit is equal to the sum of the thousands' and the tens' digit. If the digits be reversed, the resulting number will be equal to the original number increased by 4905. In what year was printing invented?

Discussion of Solutions.

3. Pr. 1. A merchant has two kinds of tea; the first is worth a cents a pound, and the second b cents a pound. How much of each kind must be taken to make a mixture of one pound worth c cents?

Let x stand for the part of a pound of the first kind, and y for the part of a pound of the second kind.

75, and

(i.) If a>c>b, the values of x and y are both positive, and the solution satisfies the conditions of the problem. Thus, if a = 100, b = c = 85, we have x, y = }.

If a<c<b, then x and y are both positive, and satisfy the conditions of the problem. That is, if the value of the pound of mixture be intermediate between the values of a pound of each of the two kinds, a definite solution is always possible.

(ii.) If c>a> b, then x will be positive and y negative. does not satisfy the conditions of the problem. Thus, if a c = 110, we obtain x, y = − z.

The solution

100, b = 75,

It is evident that a one-pound mixture of two kinds of tea which is worth more than either kind cannot be made.

(iii.) If a = b = c, then x = 8, y = 8. This solution shows that the conditions of the problem may be satisfied in an indefinite number of ways. It is evident that a one-pound mixture of two kinds of tea, that are the same in price, can be made in any number of ways, if the mixture be the same in price.

(iv.) If a = b, and ac, then x and y = ∞.

This solution does not satisfy the

conditions of the problem, since x and y must be finite proper fractions. It is also evident that a onepound mixture of two kinds of tea which are the same in price cannot be made, if the mixture is to be of a different price.

EXERCISES II.

Solve the following problems, and discuss the results:

1. If an alloy of two kinds of silver be made, and a ounces of the first be taken with b ounces of the second, the mixture will be worth m dollars an ounce. If bounces of the first be taken with a ounces of the second, the mixture will be worth n dollars an ounce. How much is an ounce of each kind of silver worth?

2. Two bodies are separated by a distance of d yards. If they move toward each other with different velocities, they will meet after m seconds: but if they both move in the same direction, the one will overtake the other after n seconds. With what velocities do the bodies move?

CHAPTER XV.

INDETERMINATE LINEAR EQUATIONS.

1. An Indeterminate Equation was defined in Ch. XIII., § 1, Art. 1, as an equation which has an indefinite number of solutions; as x + y = 5. An Indeterminate System is a system of equations which has an indefinite number of solutions.

Thus, if the system

x + y − z = 9, 2xy + 72 = 33,

be solved for x and y, we obtain

x = 142 z, y = 3 z − 5.

In these values of x and y we may assign any value to z, and obtain corresponding values of x and y.

Evidently the number of solutions will be more limited if only positive integral values of the unknown numbers are admitted.

In this chapter we shall consider a simple method of solving in positive integers linear indeterminate equations and systems.

2. Ex. Solve 4x + 7y = 94, in positive integers.

Solving for x, which has the smaller coefficient, we obtain

y must have such a value that 2 -3 y shall be divisible by 4.

tegral values of y. But since the expression

any multiple of it will be an integer. We therefore multiply its numera

an inconvenient form from which to determine in2-3 is to be an integer,

4