Question 984001
The ratio of their speeds will be {{{1:K}}} ,
with the car’s average speed being {{{K}}} times Danny's average walking speed.
Danny walks {{{x}}} miles towards home in {{{t}}} time, at his average walking speed.
His mother, covers {{{K*x}}} miles in the same {{{t}}} time,
because the car’s average speed is {{{K}}} times Danny's average walking speed.
Together, they have covered the {{{x+Kx=(K+1)x}}} miles between the station and home in {{{t}}} time.
However, there is still the drive home, which will take time {{{t}}} ,
because it is the same distance mom drove to meet Danny,
covered at the same car’s average speed.
So, the whole trip takes time {{{t+t=2t}}} .
If Danny had to walk the whole {{{(K+1)x}}} miles,
instead of just the {{{x}}} miles he walked in time {{{t}}} ,
it would take him {{{K+1}}} times longer than {{{t}}} .
It would take him {{{(K+1)t}}} time to walk the whole way home.
With his mother's help he got home in just {{{2t}}} time,
which is {{{1/3}}} of the {{{(K+1)t}}} time to walk the whole way home.
So,
{{{2t/((K+1)t)=1/3}}}-->{{{2/(K+1)=1/2}}}-->{{{K+1=2*3}}}-->{{{K+1=6}}}-->{{{K=6-1}}}-->{{{highlight(K=5)}}}