INFINITE RESULTS

439. An infinite result indicates that the problem is impossible. 1. If a man's yearly income is a dollars and his yearly expenses are a dollars, in how many years will he have saved b dollars?

That is, he will never have saved b dollars in this way.

2. A reservoir is fitted with three pipes. One pipe can fill the reservoir in 15 hours, the second can fill it in of that time, and the third pipe can empty it in 6 hours. If the reservoir is full and the three pipes are opened, in what time will it be emptied ?

That is, the reservoir will never be emptied under these conditions.

3. What number added to both terms of the fraction will make the fraction equal to 1?

Consequently, there is no such number: but the larger the number in numerical value, the nearer will the resulting fraction approach the value 1.

THE PROBLEM OF THE COURIERS

440. Two couriers, A and B, travel on the same road in the direction from X to Y at the rates of m and n miles an hour, respectively. At a certain time, say 12 o'clock, A is at P, and B is at Q, a miles from P. Find when and where they are together.

Suppose that time reckoned from 12 o'clock toward a later time is positive, and toward an earlier time, negative; also, that distances measured from P toward the right are positive, and toward the left, negative.

Let x represent the number of hours from 12 o'clock, and y the number of miles from P, when A and B are together. Then, they will be together a miles from Q.

y

Since A travels mx miles and B travels nx miles before they are together,

When a > 0 and m>n, the numerator and denominator in (3) and also in (4) are positive; hence, x and y are positive.

That is, A overtakes B sometime after 12 o'clock, somewhere at the right of P.

2. When a > 0 and m<n.

When a > 0 and m<n, both x and y are negative.

That is, at 12 o'clock B is ahead of A and gaining on him, and they were together sometime before 12 o'clock and somewhere at the left of P.

3. When a > 0 and m = n.

When a > 0 and m = n, x and y are positive and infinitely great.

That is, at 12 o'clock B is ahead of A and traveling at the same rate; consequently, he will never be overtaken by A.

4. When a = 0 and m>n or m <n.

When a = 0 and m>n or m< n, x = 0 and y = 0.

If m>n, x = +0 and y = +0.

gether, and A is passing B.

That is, at 12 o'clock A and B are to

If m < n, x = 0 and y = 0. That is, at 12 o'clock A and B are together, and B is passing A.

and y 0

=

That is, A and B are together at 12 o'clock, and since they travel at the same rate they will be together at all times.

INDETERMINATE EQUATIONS

441. While a problem that presents more unknown literal numbers than independent equations involving them is in general indeterminate (§ 214), yet frequently by the introduction of a condition or conditions not leading to equations, the number of values of the unknown numbers may be limited and these values algebraically determined. A common condition is that the results shall be positive integers.

1. Solve the equation 5x+3y= 35 in positive integers.

SOLUTION

Since x and y are positive integers, 5 x must be equal to 5 or a multiple of 5, and 3 y must be equal to 3 or a multiple of 3. Since the sum of these multiples is 35, if the multiples of 5 are subtracted from 35, one or more of the remainders will be a multiple of 3, if the problem is possible.

The only multiples of 5 that subtracted from 35 leave multiples of 3 are 5 and 4 times 5.

.. x = 1 or 4; whence, y = 10 or 5.

Or, since x must be a positive integer and by transposition 5x - 35 - 3y, the values of x must be 1, 2, 3, 4, 5, or 6, if the equation is possible. Substituting these values of x in the given equation and rejecting all those that give negative or fractional values for y, the positive integral values are found to be x = 1 or 4, and y = 10 or 5.

2. Solve the equation 5x+8y = 107 in positive integers.

form. To avoid this, the coefficient of y in the number placed equal to w

should be made equal to unity. Since 2-3 is equal to an integer, any

5

multiple of it is equal to an integer. Since 5 is contained in 3 times — 3y, — 2 y times with a remainder of y, multiplying (4) by 3,

Equations (7) and (8) are called the general solution of the given equation in integers.

To make y and x positive integers, it is evident from (7) that we must take w>0; and from (8) that we must take w <3.

Since w is an integer greater than 0 and less than 3, w = 1 or 2.

3. Determine whether the equation 10x + 15 = 53 may be satisfied by integral values of x and y.

SOLUTION. Dividing by 5, 2x + 3y = 53.

If x and y are integers, the first member is integral.

х

Since the first member is equal to the fraction 53, it cannot be an integer. Hence, x and y cannot be integers at the same time; that is, the equation is not satisfied by integral values of x and y.

Find the least integral values of x and y in the following:

21. Separate 100 into two parts one of which is a multiple of 11, and the other a multiple of 6.

22. In what ways may a weight of 19 pounds be weighed with 5-pound and 2-pound weights?

23. A man has $300 that he wishes to expend for cows and sheep. If cows cost $45 apiece and sheep $6 apiece, how many can he buy of each ?

24. If 9 apples and 5 oranges together cost 52 cents, what is the cost of one of each?