Question 175252
We don't know the distance they raced, so let's just call that {{{d}}}.  Since we know that distance equals rate times time or {{{d=rt}}}, we can also say that time equals distance divided by rate or {{{t=d/t}}}.


The question we are trying to answer is who had the shorter time.  Because Barbara averaged 9 mph for the entire course, we can describe her time {{{t[B]=d/9}}}.


Ramir's time for the first half of the race must be:  {{{t[R1]=(d/2)/8}}} and the second half:  {{{t[R2]=(d/2)/10}}}.  Then Ramir's total time must be the sum of these two:  {{{t[R]= (d/2)/8 + (d/2)/10}}} 


It is convenient in this case to use 80 as a common denominator,   {{{t[R]= 5d/80 + 4d/80 = 9d/80}}}.


Now go back and examine Barbara's time:  {{{t[B]=d/9}}}.  If we multiply by 1 in the form of {{{9/9}}}, then her time can be expressed as {{{t[B]=9d/81}}}.


If two fractions have the same numerator, the one with the larger denominator is the smaller number, hence Barbara wins.  (But just by the hair on her chinny-chin-chin -- a little more than 6 tenths of a second over a 200 meter race)