Question 588034
Add their rates of working to get their
rate working together
Let ( 1 task ) / ( t hours ) = B's rate
given:
( 1 task ) / ( t - 4 hours ) = A's rate
{{{ 1/t + 1/( t-4 ) = 1/(8/3) }}}
{{{ 1/t + 1/( t-4 ) = 3/8 }}}
Multiply both sides by {{{ t*( t-4 )*8 }}}
{{{ 8*( t-4 ) + 8t  = 3*t*( t -4 ) }}}
{{{ 8t - 32 + 8t = 3*( t^2 - 4t ) }}}
{{{ 8t -32 + 8t = 3t^2 - 12t }}}
{{{ 3t^2 - 28t + 32 = 0 }}}
Use the quadratic formula
{{{ t = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{ a = 3 }}}
{{{ b = -28 }}}
{{{ c = 32 }}}
{{{ t = (-(-28) +- sqrt( (-28)^2-4*3*32 ))/(2*3) }}}
{{{ t = ( 28 +- sqrt( 784 - 384 )) / 6 }}}
{{{ t = ( 28 +- sqrt( 400 )) / 6 }}}
{{{ t = ( 28 + 20 ) / 6 }}} ( ignore the (-) root )
{{{ t = 48/6 }}}
{{{ t = 8 }}}
It takes B 8 hrs to do the task alone
It takes A {{{ 8 - 4 }}} = 4 hrs to do the task alone
check answer:
{{{ 1/t + 1/( t-4 ) = 3/8 }}}
{{{ 1/8 + 1/( 8-4 ) = 3/8 }}}
{{{  1/8 + 1/4 = 3/8 }}}
{{{ 1/8 + 2/8 = 3/8 }}}
{{{ 3/8 = 3/8 }}}
OK