A man takes 6 hours 30 minutes in going by a cycle and coming back by scooter. He would have lost 2 hrs. and 10 minutes by going on cycle both ways. How long would it take him to go by scooter both ways?

Correct option is A

$$\text{Let the time taken to go both ways by scooter be } 2x \text{ hours.} \\\text{Then, time taken one way by scooter } = x \text{ hours} \\\text{Total time for cycle (one way) + scooter (one way)} = 6\frac{1}{2} = \frac{13}{2} \text{ hrs} \\\text{So, time taken for cycle one way } = \frac{13}{2} - x \text{ hrs} \\[10pt]\text{Now, total time for both ways by cycle is:} \\ \ \\6\frac{1}{2} + 2\frac{1}{6} = 8\frac{2}{3} = \frac{26}{3} \text{ hrs} \\ \ \\\text{So, } 2 \left( \frac{13}{2} - x \right) = \frac{26}{3} \\[10pt] \\ \ \\\Rightarrow 13 - 2x = \frac{26}{3} \\ \ \\\Rightarrow 2x = 13 - \frac{26}{3} = \frac{39 - 26}{3} = \frac{13}{3} \\ \ \\\Rightarrow x = \frac{13}{6} = 2\frac{1}{6} \text{ hrs (one way)} \\ \ \\\text{So, both ways by scooter } = 2x = \frac{13}{3} = 4\frac{1}{3} \text{ hrs} = 4 \text{ hrs } 20 \text{ min} \\[10pt]{\text{ 4 hrs 20 min}}$$​

Correct answer is (a).