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A passenger train and a freight train start toward each other at the same time from two points $300$ miles apart. If the rate of the passenger train exceeds the rate of the freight train by $15$ mph and they meet after $4$ hours, what is the rate of each train?

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Write linear functions: word problems

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Q. A passenger train and a freight train start toward each other at the same time from two points

300

miles apart. If the rate of the passenger train exceeds the rate of the freight train by

15

mph and they meet after

4

hours, what is the rate of each train?

Set up equation: Step $1$ : Set up the equation based on the total distance covered by both trains when they meet. $\newline$ Since the trains start $300$ miles apart and meet after $4$ hours, the sum of the distances they travel equals $300$ miles. Let the speed of the freight train be $f$ mph. Then, the speed of the passenger train is $f + 15$ mph. The equation based on distance is: $\newline$ $4f + 4(f + 15) = 300$
Simplify and solve: Step $2$ : Simplify and solve the equation. $\newline$ Combine like terms: $\newline$ $4f + 4f + 60 = 300$ $\newline$ $8f + 60 = 300$ $\newline$ Subtract $60$ from both sides: $\newline$ $8f = 240$ $\newline$ Divide by $8$ : $\newline$ $f = 30$
Calculate speed: Step $3$ : Calculate the speed of the passenger train. $\newline$ Since the speed of the freight train $f$ is $30$ mph, the speed of the passenger train is: $\newline$ $f + 15 = 30 + 15$ $\newline$ $f + 15 = 45$

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