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On a $130$ mile trip, a car traveled an average speed of $\newline$ $55$ mph and then reduced its speed to $40$ mph for the remainder of the trip. The trip took $2.5$ hours. For how long did the car travel at $\newline$ $40$ mph ?

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Solve quadratic equations: word problems

Full solution

Q. On a

130

mile trip, a car traveled an average speed of

\newline

55

mph and then reduced its speed to

40

mph for the remainder of the trip. The trip took

2.5

hours. For how long did the car travel at

\newline

40

mph ?

Set Up Total Time Equation: Let $t_1$ be the time the car traveled at $55$ mph and $t_2$ be the time at $40$ mph. The total time is given as $2$ . $5$ hours. Set up the equation for total time: $\newline$ $t_1 + t_2 = 2.5$
Set Up Distance Equations: Use the distance formula $D = R \times T$ for each part of the trip. The total distance is $130$ miles. Set up the equations for distances: $\newline$ $55t_1 + 40t_2 = 130$
Solve System of Equations: Solve the system of equations: $\newline$ $1$ . $t_1 + t_2 = 2.5$ $\newline$ $2$ . $55t_1 + 40t_2 = 130$ $\newline$ Multiply the first equation by $40$ : $\newline$ $40t_1 + 40t_2 = 100$ $\newline$ Subtract this from the second equation: $\newline$ $55t_1 + 40t_2 - (40t_1 + 40t_2) = 130 - 100$ $\newline$ $15t_1 = 30$ $\newline$ $t_1 = 2$
Substitute and Solve: Substitute $t_1 = 2$ back into the total time equation: $\newline$ $2 + t_2 = 2.5$ $\newline$ $t_2 = 0.5$

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