<PRE><B>Question 1125946</B>
the average speed of a plane is 8 times the average speed of a train.


let the average speed of the train be equal to R.
the average speed of the plane is therefore 8 * R.


the plane take 8 and 3/4 hours less than the train to travel 1050 kilometers.


let the time the train takes be equal to T.
then the time the plane takes is equal to T - 8.75



find the average speed of the train.


equation to use is rate * time = distance.


for the train, the equation becomes R * T = 1050


for the plane, the equation becomes 8 * R * (T - 8.75) = 1050


these are two equations that need to be solved simultaneously.


we&#39;ll use substitution.


solve for T in the first equation to get T = 1050 / R.


replace T in second equation with 1050 / R to get:


8 * R * (T - 8.75) = 1050 becomes 8 * R * (1050 / R - 8.75) = 1050.


simplify to get 8 * R * 1050 / R - 8 * R * 8.75 = 1050.


multiply both sides of this equation by R to get:


8 * R * 1050 - 8 * R * 8.75 * R = 1050 * R


combine like terms to get:


8400 * R - 70 * R^2 = 1050 * R


add 70 * R^2 to both sides of the equation and subtract 8400 * R from both sides of the equation to get:


0 = 1050 * R - 8400 * R + 70 * R^2.


combine like terms and reorder the terms in descending order of degree and flip sides to get:


70 * R^2 - 7350 * R = 0


factor to get:


70 * R * (R - 105) = 0


divide both sides of the equation by 70 to get:


R * (R - 105) = 0


solve for R to get R = 0 or R = 105.


R is obviously not 0, so R needs to be 105 kilometers per hour or not at all.


when R = 105 kilometers per hour, the equation for the train becomes 105 * T = 1050.


solve for T to get T = 10.


when R = 105 kilometers per hour, 8 * R equals 840 kilometers per hour.


the equation for the plane becomes 840 * (T - 8.75) = 1050.


since T is equal to 10, this equation becomes 840 * (10 - 8.75) = 1050.


simplify to get 840 * 1.25 = 1050 which becomes 1050 = 1050.


this confirms the solution of R = 105 and T = 10 is correct.


the problems asks for the average speed of the train.


the solution is the average speed of the train is 105 kilometers per hour.








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