A cycled from P to Q at 10 km/hr and returned at the rate of 9 km/hr. B cycled bothways at 12 km/hr. It was discovered that for the total journey B took ten minutes less than A. What is the distance between P and Q?

Let distance = d km. A's time = d/10 + d/9 = (9d+10d)/90 = 19d/90 hours. B's time = 2d/12 = d/6 hours. B took 10 minutes less: d/6 - 19d/90 = 10/60. Converting: (15d-19d)/90 = 1/6, so -4d/90 = 1/6. This gives d = -90/24 = -3.75, but taking absolute value and reconsidering the equation setup: Actually A took MORE time, so 19d/90 - d/6 = 1/6, giving 4d/90 = 15/90, so d = 15/4 = 3.75 km.