Question 1006469
Let {{{ g }}} = number of golden balls in bag at first
Let {{{ s }}} = number of silver balls in bag at first
{{{ g + s }}} = total number of balls in bag at first
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given:
(1) {{{ ( s-1 ) / ( s - 1 + g ) = 1/7 }}}
( note that {{{ s - 1 + g }}} are the number of balls remaining in bag )
(2) {{{ s / ( s + g - 2 ) = 1/5 }}}
( {{{ s + g - 2 }}} are number of balls remaining  in bag )
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Multiply both sides of (1) by {{{ 7*( s - 1 + g ) }}}
(1) {{{ 7s - 7 = s - 1 + g }}}
(1) {{{ g - 6s = -6 }}}
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Multiply both sides of (2) by {{{ 5*( s + g - 2 ) }}} 
(2) {{{ 5s = s + g - 2 }}}
(2) {{{ g - 4s = 2 }}}
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Subtract (1) from (2)
(2) {{{ g - 4s = 2 }}}
(1) {{{ -g + 6s = 6 }}}
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{{{ 2s = 8 }}}
{{{ s = 4 }}}
and
(2) {{{ g - 4s = 2 }}}
(2) {{{ g - 4*4 = 2 }}}
(2) {{{ g - 16 = 2 }}}
(2) {{{ g = 18 }}}
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{{{ 4 + 18 = 22 }}}
There are 22 balls in the bag at first
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check the answer:
Remove 1 silver ball
There are {{{ 4 - 1 = 3 }}} silver balls left
There are {{{ 22 - 1 = 21 }}} total balls left
{{{ 3/21 = 1/7 }}}
{{{ 1/7 = 1/7 }}}
OK
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Remove 2 golden balls
There are {{{ 18 - 2 = 16 }}} golden balls left
There are {{{ 4 + 16 = 20 }}} total balls left
{{{ 4/20 = 1/5 }}}
{{{ 1/5 = 1/5 }}}
OK