X and Y are two towns. A man goes from town X to town Y at an average speed of $30$ km per hour and returns at an average speed of $20$ km per hour. Find the distance between the two towns if a man takes $10$ hours on the whole.

Let the distance between the two towns be D km. Using the formula time equals distance divided by speed, the total time taken for the journey is D/30 + D/20 = 10 hours. Solving this equation by taking the least common multiple of the denominators gives 2D + 3D = 600, which simplifies to 5D = 600. Therefore, the distance between the two towns is 120 km.

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