68. From Pickleminster to Quickville. Trains $A$ and $B$ depart from Pickleminster to Quickville at the same time as trains $C$ and $D$ depart from Quickville to Pickleminster. Train $A$ meets train $C$ 120 miles from Pickleminster and train $D$ 140 miles from Pickleminster. Train $B$ meets train $C$ 126 miles from Quickville, and train $D$ at the halfway point between Pickleminster and Quickville. What is the distance from Pickleminster to Quickville? All trains travel at constant speeds, not too different from the usual.
68. There are two distances that satisfy the condition of the problem - 210 and 144 miles. The latter case is excluded, as the problem states that the trains move at speeds "not too different from the usual." (If we had taken the distance to be 144 miles, then \(A\) would have traveled 140 miles in the same time that \(B\) and \(D\) traveled 4 miles. So if the latter were traveling at 2 miles/hour, then the former would be traveling at 70 miles/hour, a speed that certainly cannot be called "not too different from the usual"!) If the distance is 210 miles, it turns out that the speeds of \(B\) and \(D\) are half the speed of \(A\), and the speed of \(C\) is \(\frac{3}{4}\) of the speed of \(A\), which seems quite reasonable.