The word problem i am having trouble completing has differen

The word problem i am having trouble completing has different distances, and that is where i get confused. Thank you so much, in advance
A cyclist travels 20 miles in the same amount of time as it takes a hiker to walk 6 miles. The rate of the cyclist is 7 mph faster than the rate of the hiker. Find the rate of the cyclist and the rate of the hiker.

Solution

EVEN THOUGH THE DISTANCES ARE DIFFERENT, THE TIMES ARE THE SAME AND THAT GIVES US THE BASIS FOR SETTING UP AN EQUATION.
Distance(d) equals Rate(r) times Time(t) or d=rt; r=d/t and t=d/r
Let r=rate of the hiker
Then r+7=rate of the cyclist
time for the hiker=6/r
time for the cyclist=20/(r+7)
And we are told that the above two times are the same. So:
6/r=20/(r+7) multiply each term by r(r+7)
6(r+7)=20r get rid of parens
6r+42=20r subtract 6r from both sides
6r-6r+42=20r-6r collect like terms
14r=42 divide both sides by 14
r=3 mph-------------------------rate of the hiker
r+7=3+7=10 mph------------------rate of the cyclist
CK
time for hiker=6/3=2 hrs
time for cyclist=20/10=2 hrs