Raghu travels from his home to college at 4 km/h and returns at 3 km/h. If the total time for the to and fro journey is 5.95 hours, what is the distance between his home and the college?

Introduction / Context:
This final question is a standard round trip problem with different speeds for going and returning. The total time for the entire journey is known, and you must find the one way distance. This reinforces the importance of treating each leg of the trip separately and then combining them using the time relationship.

Given Data / Assumptions:
Speed from home to college = 4 km/h.
Speed from college back to home = 3 km/h.
Total time for the round trip = 5.95 hours.
There are no extra delays or stops apart from the travel times at the given constant speeds.
We must find the one way distance between home and college in kilometres.

Concept / Approach:
Let the one way distance be D km. Time taken to go from home to college is D / 4 hours. Time taken to return is D / 3 hours. The sum of these times is equal to the total time 5.95 hours. This gives a single linear equation in D. Solving this equation produces the required distance.

Step-by-Step Solution:
Step 1: Let the distance between home and college be D km.Step 2: Time taken to go to college = D / 4 hours.Step 3: Time taken to return = D / 3 hours.Step 4: Total time for both journeys = D / 4 + D / 3.Step 5: According to the question, D / 4 + D / 3 = 5.95 hours.Step 6: Convert 5.95 into a fraction for cleaner calculation. 5.95 = 5 + 0.95 = 5 + 19 / 20 = (5 * 20 + 19) / 20 = 119 / 20 hours.Step 7: Write the equation as D / 4 + D / 3 = 119 / 20.Step 8: Take a common denominator for the left side: 12 is the least common multiple of 4 and 3.Step 9: D / 4 = 3D / 12 and D / 3 = 4D / 12, so D / 4 + D / 3 = 7D / 12.Step 10: Therefore 7D / 12 = 119 / 20.Step 11: Solve for D: multiply both sides by 12 to get 7D = 119 * 12 / 20.Step 12: Simplify 119 * 12 / 20 = 119 * 3 / 5 = 357 / 5.Step 13: So 7D = 357 / 5.Step 14: Divide both sides by 7: D = (357 / 5) / 7 = 357 / 35 = 51 / 5.Step 15: 51 / 5 km equals 10.2 km.

Verification / Alternative check:
If D = 10.2 km, time to college at 4 km/h is 10.2 / 4 = 2.55 hours. Time to return at 3 km/h is 10.2 / 3 = 3.4 hours. Adding these gives 2.55 + 3.4 = 5.95 hours, exactly matching the total time given in the question. Therefore the distance is consistent with all the provided information.

Why Other Options Are Wrong:
10 km would give a total time of 10 / 4 + 10 / 3 = 2.5 + 3.33..., which is not 5.95 hours.
10.5 km gives times that add to more than 6 hours, exceeding the allowed total time.
20 km doubles the distance and leads to a much larger total time than 5.95 hours.
9.5 km gives too small a total time when the speeds are applied, so it also fails the condition.

Common Pitfalls:
Some students mistakenly use average speed instead of working with separate legs of the journey. Others round 5.95 too early, which can introduce errors. Always keep the time value accurate, work symbolically with fractions when possible, and substitute back to confirm your result.

Final Answer:
The distance between Raghu home and his college is 10.2 km.