Question 1006498
Let {{{ t[1] }}} = time in minutes for the larger pipe
to fill cistern when filling alone
Rate of filling = {{{ 1/t[1] }}}
Let {{{ t[2] }}} = time in minutes for both pipes to
fill cistern when filling together
---------------------------
{{{ t[1] + 24 }}} = time in minutes for smaller pipe to 
fill cistern when filling alone
Rate of filling = {{{ 1/( t[1] + 24 ) }}}
{{{ t[2] + 32 }}} = this is also time in minutes for smaller pipe to 
fill cistern when filling alone
----------------------------
{{{ t[1] + 24 = t[2] + 32 }}}
{{{ t[2] = t[1] - 8 }}}
-----------------------------
Add their rates of filling alone to get rate filling together
{{{ 1/t[1] + 1/( t[1] + 24 ) = 1/( t[1] - 8 ) }}}
Multiply both sides by {{{ t[1]*( t[1] + 24 )*( t[1] - 8 ) }}}
---------------------------------------------
{{{ ( t[1] + 24 )*( t[1] - 8 ) + t[1]*( t[1] - 8 ) = t[1]*( t[1] + 24 ) }}}
{{{ t[1]^2 + 24t[1] - 8t[1] - 192 + t[1]^2 - 8t[1] = t[1]^2 + 24t[1] }}}
{{{ t[1]^2 - 16t[1]  = 192 }}}
Complete the square
{{{  t[1]^2 - 16t[1] + (16/2)^2 = 192 + (16/2)^2 }}}
{{{  t[1]^2 - 16t[1] + 64 = 192 + 64 }}}
{{{ ( t[1] - 8 )^2 = 256 }}}
{{{ ( t[1] - 8 )^2 = 16^2 }}}
{{{ t[1] - 8 = 16 }}}
{{{ t[1] = 24 }}}
The larger pipe takes 24 min filling alone
check:
{{{ 1/t[1] + 1/( t[1] + 24 ) = 1/( t[1] - 8 ) }}}
{{{ 1/24 + 1/( 24 + 24 ) = 1/( 24 - 8 ) }}}
{{{ 1/24 + 1/48 = 1/16 }}}
{{{ 2/48 + 1/48 = 3/48 }}}
{{{ 3/48 = 3/48 }}}
OK