Answer
$2.7$ miles.
Work Step by Step
Since we have speed in miles/HOUR, convert the times from minutes to hours.
$20$ min = $\displaystyle \frac{20}{60}= \displaystyle \frac{1}{3}$ hours,
$45$ min =$\displaystyle \frac{45}{60}= \displaystyle \frac{3}{4}$ hours.
Let the average speed on his bike be $x$ mph.
The speed by car is then $(x+4.5)$ mph.
Distance traveled in any of the two legs = (average speed)(time of travel).
By car, distance = $(x+4.5)\displaystyle \cdot\frac{1}{3}$
By bike, distance = $x\displaystyle \cdot\frac{3}{4}.$
The two distances are equal, and we have the equation:
$(x+4.5)\displaystyle \cdot\frac{1}{3}=x\cdot\frac{3}{4}\qquad$ ... multiply both sides by 12 (LCM of 3 and 4)
$(x+4.5)\displaystyle \cdot\frac{1}{3}\cdot 12=x\cdot\frac{3}{4}\cdot 12$
$(x+4.5)\cdot 4=9x\qquad$ ... distribute and simplify
$ 4x+18=9x\qquad$ ... subtract 4x
$ 18=5x\qquad$ ... divide by 5
$x=\displaystyle \frac{18}{5}=3.6$ mph (by bike)
By bike, distance = $x\displaystyle \cdot\frac{3}{4}=3.6\cdot\frac{3}{4}=2.7$ miles.
