A person running an 800 meter race average 130 meters per minute for the first  $\frac{3}{4}$ of the race. The average speed for the remainder of the race is 145 meters per minute. What is the 3 person's average speed for the entire 800 meters rounded so the nearest whole numbered?

Let, the speed of the train during returning journey be x kmph 

Speed during onward journey=  $(x+ 25\% of x)=\left(x+ \frac{25x}{100}\right)=\frac{5x}{4}$ kmph 

Distance covered in onward journey = $\frac{800}{2}=400 km$

Time taken by the train on onward journey= $\frac{{\displaystyle \frac{400}{5x}}}{4}$ hours [As, time= Distance/Speed]

And time by the train in returning journey = $\frac{400}{x}$ hours

According to the question,

$$\begin{gathered} \frac{{\displaystyle \frac{400}{5x}}}{4}+\frac{400}{x} =16 \\ \Rightarrow \left(\frac{400}{1}\times \frac{4}{5x}\right)+\frac{400}{x}=16 \\ \Rightarrow \frac{320}{x}+\frac{400}{x}=16 \\ \Rightarrow \frac{320+400}{x}=16 \\ \Rightarrow 16x=720 \\ \therefore x=45 kmph \end{gathered}$$

Speed of the train in the onward journey=$\frac{5x}{4}=\frac{5\times 45}{4}=56.25$ kmph 

প্রশ্নে বলা হচ্ছে, 800 মিটার একটি দৌড় প্রতিযোগিতায় এক ব্যক্তি $\frac{3}{4}$অংশ পথ 130 মিটার/ মিনিট এবং পরবর্তী পথ 145 মিটার/ মিনিট বেগে গেলে মোট দৌড়ে তার গড় গতিবেগ কত ছিল?

Here, first part of race = $800\times \frac{1}{2}$= 400 meter

∴ Second part of race = 800-400 = 400 meter

Time taken to cross first half of race

$=\frac{400}{130}$ $\left[As, v =\frac{d}{t}\right]$

$=\frac{40}{13}$ meter/minute

Again, Time taken to cross second part of race = $\frac{400}{145}=\frac{80}{29}$meter/minute

∴ Average speed $=\frac{800}{{\displaystyle \frac{40}{13}}+{\displaystyle \frac{80}{29}}}=\frac{800}{3.07 + 2.76}$

$=\frac{800}{5.83}=\frac{800}{6}$ [nearest whole of 5.83 is 6]

= 133.333 meter per minute

= 133 meter per minute (Approximately)

মোট সময় নেয় ৩ ঘণ্টা

যাত্রা বিরতি ১ ঘণ্টা

নিট সময় নেয় = ৭ - ১ = ৬ ঘণ্টা

মোট দূরত্ব = ৩৩০ কি.মি

∴ গড় গতিবেগ $\frac{৩৩০}{৬}$ কি.মি/ ঘণ্টা = ৫৫ কি.মি / ঘণ্টা