Question 831367
This general type of travel problem seems to be becoming common.


A jet plane, flying v mph faster than a propeller plane, travels d miles in p hours less time than the propeller plane takes to fly the same distance. How fast does each plane fly?   Assume v and p are positive numbers.


Additionally, let r be the speed of the propeller plane; and let h be the time for the propeller plane to make the d distance.


_______________speed___________time________________distance
JET____________r+v____________h-p__________________d
PROP____________r______________h___________________d


The UNKNOWN variables are r and h.  Use the uniform rates equation for travel, R*T=D, for Rate, Time, Distance.


EQUATIONS TO BEGIN:
{{{(r+v)(h-p)=d}}}, and {{{rh=d}}};


The steps will very neatly give a useful equation...
...starting with the jet's equation.
{{{rh+vh-rp-vp=d}}}
Notice that we already have the propeller plane's equation rh=d allowing a substitution:
{{{d+vh-rp-vp=d}}}
{{{vh-pr-pv=0}}}
We can again use the propeller's simple equation, as {{{h=d/r}}}, and then get
{{{v(d/r)-pr-pv=0}}}
{{{vd-pr^2-pvr=0}}}
{{{-pr^2-pvr+vd=0}}}
{{{highlight(pr^2+pvr-vd=0)}}}-----QUADRATIC EQUATION IN THE ONE VARIABLE, r.  Use general solution to a quadratic equation to solve for r.  Use found value to get the value for h.