d=r*t

16=(r+2.78)*t (the winner was 2.78 mph faster)

16=(r)*(t+1) ( the loser took 1 hour longer to finish)

they are both equal to 16 so set them equal to each other

(r+2.78)*t=(r)*(t+1)

rt+2.78t=rt+r

subtract rt from both sides

2.78t=r

substitute into either equation; I used the first

(2.78t+2.78)*t=16

2.78t^{2}+2.78t-16=0

use the quadratic equation

[(-2.78+(√(2.78^{2}+4*2.78*16)]/5.56

[(-2.78+√7.7284+177.92)]/5.56

[(-2.78+√185.648)]/5.56

(-2.78+13.6252)/5.56

10.84528/5.56

1.95 hr=t

t=1.95 hr for the winner

t=1.95+1=2.95 hr for the loser

16=(r+2.78)*1.95

16=1.95r+5.421

16-5.421=1.95r

10.579=1.95r

10.579/1.95=r

5.425=r

r=5.425 mph for the loser

r=5.425+2.78=8.205 mph for the winner

check:

16=8.205*1.95=15.9997 for the winner

16=5.425*2.95=16.0037 for the loser