Question 758663
A man walked 12km at a certain rate and then 6km farther at a rate of
 1 and a half km/hour faster.
If he had walked the whole distance at the faster rate,
 his time would have been 20 minutes less.
How long did it really take him to walk the 18km?
:
let r = "a certain rate"
then
(r+1.5) = the faster rate
:
Change 20 min to {{{1/3}}} hr
:
Actual time - faster time = 20 min
{{{12/r}}} + {{{6/((r+1.5))}}} - {{{18/((r+1.5))}}} = {{{1/3}}}
Subtract like terms
{{{12/r}}} - {{{12/((r+1.5))}}} = {{{1/3}}}
multiply by  3r(r+1.5)
3(r+1.5)(12) - 3r(12) = r(r+1.5)
(3r + 4.5)(12) - 36r = r^2 + 1.5r
36r + 54 - 36r = r^2 + 1.5r
0 = r^2 + 1.5r - 54
Use the quadratic formula to find r
{{{r = (-1.5 +- sqrt(1.5^2-4*1*-54 ))/(2*1) }}}
You can do the math, I got a positive value: r = 6.6366 km/hr
:
Find the time:
t(r) = {{{12/6.6366}}} + {{{6/((6.6366+1.5))}}}
t(r) = 1.8 + .74 
t(r) = 2.545 hrs
:
:
You can check this by finding the time at the faster rate
{{{18/(6.6366+1.5)}}}  Should be .333 hrs less