Question 868549
{{{L}}}= length of the playground, in feet
When Roman and Jems first meet and pass each other 30ft from the east end,
Jems has covered {{{30}}} feet, and Roman has covered {{{L-30}}} feet.
When Roman and Jems meet again at {{{14}}} ft from the west end,
Jems has covered {{{L+14}}} feet, while Roman is {{{14}}} feet short of 2 playground lengths,
meaning that Roman has covered {{{2L-14}}} feet.
Since each of them rode at a constant speed,
the ratio of their speeds and the ratio of the distances covered each time they meet is always the same,
so {{{(2L-14)/(L+14)=(L-30)/30}}} .
We solve that equation to get {{{L}}} :
{{{(2L-14)/(L+14)=(L-30)/30}}}
{{{30(2L-14)=(L+14)(L-30)}}}
{{{60L-420=L^2+14L-30L-420}}}
{{{60L=L^2+14L-30L}}}
{{{L^2+14L-30L-60L=0}}}
{{{L^2-76L=0}}}
{{{L(L-76)=0}}}
{{{L=0}}} is a solution of that equation, but it does not make sense.
{{{highlight(L=76)}}} is the solution that makes sense.
The length of the playground is {{{highlight(76)}}} feet.