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. quartered on each floor?

How many men are

If 198 be added to those of the given

62. The sum of the three digits of a number is 9. the number, the digits of the resulting number are 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.

(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 = 75, and C = 85, we have x = f, y = f.

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

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.

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.

or

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

an inconvenient form from which to determine in-
2-3y
4

Iis to be an integer,

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

tor by the least number which will make the coefficient of y one more than a multiple of the denominator, i.e., by 3. We then have

6-99-1 2y+2, an integer.

4

Therefore, as above,

Whence

Then, from (1) and (2),

4

=m, an integer.

y = 2-4 m.

x= 20 + 7 m.

Any integral value of m will give to x and y integral values.

Therefore the only admissible values of m are 0, −1, − 2.

(2)

(3)

In solving a system of two linear equations in three unknown numbers, we first eliminate one of the unknown numbers, and apply to the resulting equation the preceding method.

3. Pr. A party of 20 people, consisting of men, women, and children, pay a hotel bill of $67. Each man pays $5, each woman $4, and each child $1.50. How many of the company are men, how many women,

Since x, y, and z are to be positive, we have:

from (4), m<1; from (5), m>— 2.

(4)

(5)

(6)

Consequently the company may have consisted of 2 men, 12 women, 6 children; or of 7 men, 5 women, and 8 children.

4. Not every linear indeterminate equation can be solved in positive integers.

The general form of such an equation is evidently ax + by = c, wherein a, b, and c are integers.

If a and b have a common factor, ƒ say, then a = fa', b = fb', wherein a' and b' are integers. The equation may then be written

Since a' and b' are integers, must be an integer, if x and y are to have integral values; that is, ƒ must be also a factor of c. Therefore,

(i.) The linear indeterminate equation ax + by = c cannot be solved in positive integers if a and b have a common factor, which is not a factor of c.

E.g., 2x-4y = 5 cannot be solved in positive integers.

We can infer at once that

(ii.) If a and b are positive and c negative, the equation ax + by = c cannot be solved in positive integers.

For ax + by would then be positive and could not be equal to c, a negative number.

E.g., 2x+5y = 6 cannot be solved in positive integers.

The case in which a and b are negative and c positive is evidently included in (ii.).

We therefore conclude that

(iii.) The linear indeterminate equation ax + by = c can be solved in positive integers only when a, b, and c are all positive, or when a and b have opposite signs; and when, in both cases, a and b do not have a common factor which is not also a factor of c.