man travels 400km partly by train …

Sol: Let the speed of the train = x kmph and that of the car = y kmph.

Total distance = 400 kms.

Time 1 = t₁ = 4 hrs 30 mins = 4.5 hrs.
Time 2 = t₂ = 4 hrs 30 mins + 15 mins = 4 hrs 45 mins = 4.75 hrs.
Physics formula used = Distance = speed x time.
      Case 1:
  If he covers half the distance by train and the rest of by the car then he travels 200 kms by each.
 total time taken = distance/speed of train + distance/speed of car
{tex}\huge {=> \ 4.5= \frac {200}{x} + \frac {200}{y}}\\ \huge {=> \ \frac{4.5}{200}= \frac {1}{x} + \frac {1}{y}}\\ \text{Let u = }{\frac1x} \text{ and v = }{\frac1y}\\ \huge {=> \ \{\frac{4.5}{200}= u+v}\}.............eq.1\\{/tex}

Case 2:
 if he travels 100km by train and rest by the car, then he travels 300 kms by car.
 total time taken = distance/speed of train + distance/speed of car
{tex}\huge {=> \ 4.75= \frac {100}{x} + \frac {300}{y}}\\ \huge {=> \ \{\frac{4.75}{100}= u + 3v}\}.............eq.2\\{/tex}

Subtracting eq. 2 from eq. 1, we will get:

{tex}\huge {=> \ \frac{4.75}{100}-\frac{4.5}{200}= 2v}\\ \huge {=> \ \frac{1}{40}= 2v}\\ \huge {=> \ v=\frac{1}{80}}\\ \text{Therefore, putting value of v in eq. 1 we will get:}\\ \huge {=> \ u=\frac{1}{100}}\\{/tex}

{tex}\huge {=> \ u=\frac1x=\frac{1}{100}}\\ .\\ \text{Therefore, } \large \boxed{x=100} \ \ and,\\ \huge {=> \ v=\frac1x=\frac{1}{80}}\\ .\\ \text{Therefore, } \large \boxed{y=80}\\ \text{Therefore, } \large \text{ speed of the train = 100 kmph and speed of the car = 80 kmph.}\\ {/tex}