Question 436704
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A plane flies 1500 miles against the wind in 3 hours and 45 minutes. 
The return trip with the wind takes 3 hours. 
Assume that the wind speed stays constant. 
Find the speed of the wind and the speed of the airplane with no wind.
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        @mananth solved the problem making tons of unnecessary calculations.

        Therefore, his solution can scary a reader and is a bad way to teach.

        I will show below a standard solution approach to make minimum calculations.



<pre>
The flight against the wind took 3 hours and 45 minutes = 3{{{3/4}}} hours = {{{15/4}}} hours.

The effective rate was  {{{1500/((15/4))}}} = {{{(1500*4)/15}}} = 100*4 = 400 miles per hour.


It gives you first equation 

    u - v = 400  mph  (1),

since the effective rate against the wind is the difference of the airspeed u and the rate of the wind v.



The flight with the wind took 3 hours.

The effective rate was  {{{1500/3}}} = 500 miles per hour.


It gives you second equation 

    u + v = 500  mph  (2),

since the effective rate with the wind is the sum of the airspeed u and the rate of the wind v.


Now you have a system of two equations for u and v.


To find u, add equations (1) and (2).  The terms with 'v' cancel each other, and you will get

    2u = 400 + 500 = 900,  u = 900/2 = 450.


Now from equation (2)  v = 500 - 450 = 50.


Thus the problem is solved, and the <U>ANSWER</U> is

     airspeed is 450 mph  (the rate in still air, or relative the air), and

     the rate of the wind is 50 mph.
</pre>

Solved, making the minimum calculations - making only those calculations that are really needed.


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Now, after seeing my solution, it should be clear to you that the answer is the nice round numbers,
while ugly numbers from @mananth are (1) incorrect and (2) are the consequences of his 
"approximate" calculations that only harm the solution.



Actually, the input numbers in this problem are "very nice" and lead to the nice
answer, if to treat the problem properly.


The way as I treat the problem in my post is the EXPECTED way on how the problem SHOULD be treated.