Question 810752
Let {{{ x }}} = the number staying overnight
{{{ 4800 / x }}} = the bill for each person
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{{{ 4800 = ( 4800/( x+4 ) + 200 )*x }}}
( total bill ) = ( same bill divided among ( x+4 ) people + add the savings back in for each person )
times ( original number of people )
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{{{ 4800 = 4800x / ( x + 4 ) + 200x }}}
{{{ 4800*( x+4 )  = 4800x + 200x*( x+4 ) }}}
{{{ 4800x + 19200 = 4800x + 200x^2 + 800x }}}
{{{ 200x^2 + 800x - 19200 = 0 }}}
{{{ x^2 + 4x - 96 = 0 }}}
{{{ ( x + 12 )*( x - 8 ) }}} ( by inspection )
{{{ x = 8 }}}
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There are 8 people staying overnight
check:
{{{ 4800/8 = 600 }}} per person
With 4 more people:
{{{ ( 600 - 200 )*12 = 4800 }}}
OK