Question 1011058
This form of travel rates problem is common as an exercise.

<pre>
           rate     time      distance
IF         r+6      t-1/4      390
Usual      r        t          390
</pre>
<pre>
           rate     time      distance
IF         r+k      t-h        d
Usual      r        t          d
</pre>
Be sure you understand what is happening in and between the two data tables above.


{{{system((r+k)(t-h)=d,rt=d)}}}


{{{rt+kt-hr-hk=d}}}
{{{d+kt-hr-hk=d}}}
{{{kt-hr-hk=0}}}
{{{kt=hr-hk}}}
{{{t=(hr-hk)/k}}}------formula for t in terms of r


{{{rt=d}}}
{{{r=d/t}}}
{{{r=d*(1/t)}}}
{{{r=d(k/(hr-hk))}}}
{{{r(hr-hk)=dk}}}
{{{highlight_green(hr^2-hkr-dk=0)}}}-----quadratic equation in the unknown variable, r.
You can use the formula for general solution of a quadratic equation and try to simplify
that, if at all possible,  ... or you can simply substitute the given, known values for h
, k, and d, now and then deal with the resulting form of quadratic equation.


{{{(1/4)r^2-(1/4)*6r-390*6=0}}}
{{{(1/4)r^2-(3/2)r-2340=0}}}
{{{highlight_green(r^2-6r-9360=0)}}}----now the quadratic equation is less general, fewer symbols.


{{{r=(6+- sqrt(36+4*9360))/2}}}


{{{r=(6+- sqrt(37476))/2}}}
{{{r=(6+- sqrt(4*9*3*347))/2}}}
{{{r=(6+- 2*3sqrt(3*347))/2}}}----you need the PLUS form.
{{{highlight(r=3+3sqrt(3*347))}}}
Do whatever you need with this value.


--
347 and 3 both are prime numbers.  Decimal approximate for the solution is {{{highlight(r=99.8)}}}.