Question 874131
ONE WAY TO DO IT:
In the time that Ann ran {{{5}}} miles,
Betty only covered {{{5-1/4=20/4-1/4=19/4}}} miles.
Betty is slower than Ann.
Compared to Ann's speed, Betty's speed is
{{{(19/4)/5=19/20}}} of Ann's speed.
Betty's speed is {{{19/20}}} of Ann's speed.
 
In the time that Betty ran {{{5}}} miles,
Cindy only covered {{{5-1/3=15/3-1/3=14/3}}} miles.
Cindy is slower than Betty .
Compared to Betty's speed, Cindy's speed is
{{{(14/3)/5=14/15}}} of Betty's speed.
Cindy's speed is {{{14/15}}} of Betty's speed.
 
Compared to Ann's speed, Cindy's speed is
{{{14/15}}} of {{{19/20}}} of Ann's speed, or
{{{(14/15)(19/20)=14*19/(15*20)=7*19/(15*10)=133/150}}} of Ann's speed.
So, in the time that Ann ran  {{{5}}} miles,
Cindy covered {{{5*(133/150)=133/30}}} miles.
That means that when Ann finished the race,
Cindy had only ran {{{133/30}}} out of the {{{5}}} miles.
So Cindy was
{{{5-133/30=150/30-133/30=17/30=about0.57}}} miles back.
 
ANOTHER WAY:
We could say that Ann ran the 5 miles in {{{t}}} minutes, and then we can calculate Ann's, Betty's and Cindy"s speeds in miles per minute based on that:
{{{5/t}}}= Ann's speed.
Since Betty was {{{1/4}}} mile back,
and had only covered {{{19/4}}} miles in that time {{{t}}},
{{{(19/4)/t=19/4t}}}= Betty's speed.
So Betty would take {{{5/(19/4t)=5(4t/19)=20t/19}}} minutes to finish.
In that time, Cindy had only ran {{{5-1/3=14/3}}} miles,
so Cindy's speed, in miles per minute, is
{{{(14/3)/(20t/19)=(14/3)(19/20t)=14*19/(3*20t)=266/60t=133/30t}}} .
That means that, in the time {{{t}}} that it took for Ann to finish the race,
Cindy had covered {{{t(133/30t)=133/30}}} miles.
So, to get to the finish line, Cindy still had to run
{{{5-133/30=150/30-133/30=17/30}}} miles back.
 
SIMPLER:
We can set an arbitrary time.
Let's say that Ann's time was 40 minutes.
Ann ran at {{{5/40=0.125}}} miles per minute.
Betty had ran {{{5-1/4=4.75}}} miles in those 40 minutes.
Betty ran at {{{4.75/40=0.11875}}} miles per minute.
At that speed it would take her
{{{5/0.11875=about 42.105}}} minutes to finish the 5 mile race.
In that time, Cindy had only run {{{5-1/3=4&2/3=about4.667}}} miles.
Cindy's speed was about {{{4.667/42.105=about0.11084}}} miles per minute.
At that speed, in the 40 minutes that it took Ann to finish the race,
Cindy covered approximately {{{40*0.11084=4.4336}}} miles,
and at that point she was about
{{{5-4.4336=0.5664}}} miles back.