18hun699
contestada

Solve the following word problem. A man travels from Town X to Town Y at an average rate of 50 mph and returns at an average rate of 40 mph. He takes a 1/2 hour longer than he would take if he made the round trip at an average of 45 mph. What is the distance from Town X to Town Y?

Respuesta :

wizard123
distance = rate*time

Let distance from X to Y be 'd' 

From the equation we can solve for 'time'

time = distance/rate

The total time it takes to go 50 mph ,then 40 mph is:

[tex]\frac{d}{50} + \frac{d}{40}[/tex]

The total time it takes to go 45 mph round trip is:

[tex]\frac{2d}{45} [/tex]

We know that the round-trip at 45 mph takes half-hour shorter, so by adding '1/2' to its time it will be equal to the time to go 50 then 40.

[tex]\frac{2d}{45} + \frac{1}{2} = \frac{d}{50} + \frac{d}{40} \\ \\ \frac{4d+45}{90} = \frac{9d}{200} \\ \\ 800d+9000 = 810d \\ \\ 10d = 9000 \\ \\ d = 900[/tex]

Answer: The distance from X to Y is 900 miles.

The distance from Town X to Town Y is 900 miles.

Given to us,

  • Average speed from Town X to Town Y = 50 mph,
  • Average speed from Town Y to Town X = 40 mph,
  • Average speed for round trip = 45 mph,
  • Time taken is  [tex]\dfrac{1}{2}[/tex] hour more if taken the round trip.

Assumption

Let the distance between town X and town Y be x.

Time Calculation

Time is taken for traveling from Town X to Town Y  [tex]= \dfrac{Distance}{Speed}= \dfrac{x}{50}[/tex]

Time is taken for traveling from Town Y to Town X  [tex]= \dfrac{Distance}{Speed}= \dfrac{x}{40}[/tex]

Total time = Time is taken for traveling from Town X to Town Y + Time is taken for traveling from Town Y to Town X

Total time =  [tex]\dfrac{x}{50}+\dfrac{x}{40}[/tex]

Time is taken for a round trip at an average speed of 45 mph[tex]= \dfrac{Distance}{Speed}= \dfrac{2x}{45}[/tex]

Comparison of Time

[tex]\dfrac{x}{50}+\dfrac{x}{40}= \dfrac{1}{2}+ \dfrac{2x}{45}[/tex]

[tex]\dfrac{4x}{200}+\dfrac{5x}{200}= \dfrac{45}{90}+ \dfrac{4x}{90}\\\\\dfrac{4x+5x}{200}= \dfrac{45+4x}{90}\\\\\dfrac{9x}{200}= \dfrac{45+4x}{90}\\\\810x = 9000+800x\\810x-800x = 9000\\10x = 9000\\\\x=\dfrac{9000}{10}\\\\x=900[/tex]

Hence, the distance from Town X to Town Y is 900 miles.