Let x represent the number of years that will elapse.
Then 40 + x = 5 (24 + x); 40 + x = 120 + 5x, or x = - 20.
According to this analysis, A will be five times as old as B in — 20 years. The meaning of this result ought to be at once evident to a thoughtful student. Were the result any positive number of years, we would simply count that number forward from the present time (represented by the word " is " in the problem) ; manifestly then the — 20 years refers to past time. Hence the problem should read, "A is 40 years old, and B's age is three-fifths of A's. When was A five times as old as B ?"
Suppose the problem read
A is 40 years old, and B's age is three-fifths of A's : find the time at which A's age is five times that of B.
Let us assume that x years will elapse.
Then 40 + x = 5(24 + x); x = -20.
Interpreting this result, we see that we should have assumed that x years had elapsed.
The student will notice that the word " will " in the first statement suggested that we should assume x as the number of years that would elapse, and that the negative result showed a fault in the enunciation of the problem ; but that the problem, as given in the next discussion, permitted us to make one of two possible suppositions as to the nature of the unknown quantity, so that the negative result indicates simply a wrong choice.
Hence in the solution of problems involving equations of the first degree, negative residts indicate
(1) A fault in the enunciation of the problem, or
(2) A wrong choice between two possible suppositions, as to the nature of the unknown quantity, allowed by the problem.
Generally it will be easy for the student to make such changes as will give an analogous possible problem.
