Question

Ian and Jeff leave their apartment to go to a football game 54 miles away. Ian drives his car 24 mph faster than Jeff can ride his bike. If it takes Jeff 3 hours longer than Ian to get to the game, what is Jeff's speed?

Answer 1

Jeff's speed is 12 mph

EXPLANATION :

The football game is 54 miles from their apartment.

Ian drives 24 mph faster than Jeff,

so if Jeff's speed is x,

Ian's speed will be x + 24

It takes 3 hours longer than Ian to get to the game,

so if Ian takes y hours to get to the game,

Jeff's time will be y + 3

Recall the equation of speed, time and distance.

`d=rt`

where d = distance

r = speed

t = time

Equation for Jeff :

`54=x(y+3)`

Equation for Ian :

`54=(x+24)(y)`

Express Jeff's equation as x in terms of y.

`x=(54)/(y+3)`

Substitute this to Ian's equation :

`\begin{gathered} 54=(x+24)(y) \\ 54=((54)/(y+3)+24)(y) \\ 54=(54y)/(y+3)+24y \\ \text{ Cross multiply :} \\ 54(y+3)=54y+24y(y+3) \\ 54y+162=54y+24y^2+72y \\ 0=24y^2+54y+72y-54y-162 \\ 0=24y^2+72y-162 \\ \text{ Divide the equation by 6 :} \\ 0=4y^2+12y-27 \\ 4y^2+12y-27=0 \\ (2x-3)(2x+9)=0 \end{gathered}`

Equate the factors to 0.

2x - 3 = 0

2x = 3

x = 3/2

2x + 9 = 0

2x = -9

x = -9/2

Note that there's no negative time, so we will consider x = 3/2 only

Substitute x = 3/2 to Jeff's equation :

`\begin{gathered} 54=x(y+3) \\ 54=x((3)/(2)+3) \\ 54=x((9)/(2)) \\ 108=9x \\ x=(108)/(9) \\ x=12 \end{gathered}`