Average rate

Question

A car and a bus both completed a 240 mile trip. The car averaged 10mph faster than the bus  and completed the trip in 48 minutes less time than the bus. Find the average rate in miles per hour of the bus.

Answer 1

let x = speed of car

 

then x - 10 = speed of bus

 

Distance = (Rate)(Time), then Time = distance /rate

 

Time of car = Time of bus - 0.8    (48 minutes = 48/60 hr = 0.8 hr)

 

So, 240/x = 240/(x-10) - 4/5

 

Multiply by the LCD, 5x(x-10), to get:  

 

1200(x-10)= 1200x - 4x(x-10)

 

Divide by 4:  300(x-10) = 300x -x²+ 10x

 

x² - 10x - 3000 = 0

 

(x-60)(x+50) = 0

 

x = 60 or -50

 

Since x can't be negative, x = 60 miles per hour

 

Speed of car = 60 mph

Speed of bus = 50 mph

 

 

Answer 2

We use the formula

 

v = d / t

 

where:

v = speed (in mph)

d = distance (in miles)

t = time (in hours)

 

Write down what is known:

d_car = d_bus = 240

 

t_car = t_bus - 0.8

 

 

v_car - v_bus = 10

 

 

(d_car / t_car) - (d_bus / t_bus)  = 10

 

 

[240 / (t_bus - 0.8)] - [240 / t_bus] = 10

 

 

Now we have an equation in terms of t_bus.  Solve for t_bus. 

 

[240t_bus - 240(t_bus - 0.8)] / [t_bus (t_bus - 0.8)] = [10t_bus (t_bus - 0.8)] / [t_bus (t_bus - 0.8)]

 

 

Equate the numerators.  We let  x=t_bus.

 

240x - 240(x - 0.8) = 10x(x - 0.8)

 

 

240x - 240x + 192 = 10x² - 8

 

192 = 10x² - 8

 

200 = 10x²

 

20 = x²

 

4.472 = x

 

 

This means that it takes the bus 4.742 hours.  Divide 240 by this value to get the speed of the bus.

 

speed of bus = 240 miles / 4.472 hr

                   = 53.67 mph