Ram started walking from his home towards his office with a uniform speed so that he could reach his office exactly on time. After walking for some time, he realised that he had left an important file at home. He immediately turned around and started walking towards his home with the same speed. He reached home, picked up the file in no time and started riding a bicycle towards his office at such a speed that he would reach his office exactly on time. The ratio of Ram’s cycling speed to his walking speed is 5 : 3. The percentage of the distance between his home and office that Ram walked towards his office is ___________ .

To solve this problem, let's break it down step by step.

1. Define Variables:

   • Let the distance between Ram's home and office be $D$.
   • Let Ram's walking speed be $W$.
   • Let Ram's cycling speed be $C$.
   • Let the time it takes for Ram to walk from his home to his office be $T$.

2. Given Information:

   • The ratio of Ram’s cycling speed to his walking speed is $\frac{C}{W} = \frac{5}{3}$.
   • Therefore, $C = \frac{5}{3}W$.

3. Time Calculation:

   • The time it takes for Ram to walk the entire distance $D$ is $T = \frac{D}{W}$.

4. Walking and Returning:

   • Let the distance Ram walked before realizing he forgot the file be $d$.
   • The time taken to walk this distance is $\frac{d}{W}$.
   • The time taken to return home is also $\frac{d}{W}$.

5. Total Time Spent Walking:

   • The total time spent walking is $2 \times \frac{d}{W} = \frac{2d}{W}$.

6. Cycling to Office:

   • The remaining distance to the office after returning home is $D$.
   • The time taken to cycle this distance is $\frac{D}{C} = \frac{D}{\frac{5}{3}W} = \frac{3D}{5W}$.

7. Total Time Calculation:

   • The total time spent walking and cycling should equal the time it would have taken to walk the entire distance $D$.
   • Therefore, $\frac{2d}{W} + \frac{3D}{5W} = \frac{D}{W}$.

8. Solve for $d$:

   • Multiply through by $W$ to clear the denominators: $2d + \frac{3D}{5} = D$
   • Rearrange to solve for $d$: $2d = D - \frac{3D}{5}$ $2d = \frac{5D}{5} - \frac{3D}{5}$ $2d = \frac{2D}{5}$ $d = \frac{D}{5}$

9. Percentage Calculation:

   • The percentage of the distance between his home and office that Ram walked towards his office is: $\left( \frac{d}{D} \right) \times 100 = \left( \frac{\frac{D}{5}}{D} \right) \times 100 = \frac{1}{5} \times 100 = 20\%$

Therefore, the percentage of the distance between his home and office that Ram walked towards his office is 20%.