Question 703267
Let {{{ a }}} = Ann's amount at the start
Let {{{ b }}} = Betty's amount at the start
Let {{{ c }}} = Carol's amount at the start
They ended up with $8 each, and {{{ 3*8 = 24 }}},
so, at the start,
{{{ a + b + c = 24 }}}
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Ann gives Betty {{{ b }}} dollars and gives Carol {{{ c }}} dollars
Ann is left with {{{ a - b - c }}}
Betty has {{{ 2b }}}
Carol has {{{ 2c }}}
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Betty does the same thing as Ann did
She gives Ann {{{ a - b - c }}}, so Ann now has {{{ 2a - 2b - 2c }}}
She gives Carol {{{ 2c }}}, so now Carol has {{{ 4c }}}
Betty is left with {{{ 2b - ( a - b - c ) - 2c }}} or
{{{ 2b - a + b + c - 2c }}}
{{{ -a + 3b - c }}}
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Now Carol does the same thing
She gives Ann {{{ 2a - 2b - 2c }}} and Ann now has {{{ 4a - 4b - 4c }}}
She gives Betty {{{ -a + 3b - c }}} and Betty now has {{{ -2a + 6b - 2c }}}
Carol is left with {{{ 4c -( -a + 3b - c ) - ( 2a - 2b - 2c ) }}}, or
{{{ 4c + a - 3b + c - 2a + 2b + 2c }}}
{{{ -a - b + 7c }}}
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Each woman ends up with $8, so
Ann:
(1) {{{ 4a - 4b - 4c = 8 }}}
Betty:
(2) {{{ -2a + 6b - 2c = 8 }}}
Carol:
(3) {{{ -a - b + 7c = 8 }}}
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(1) {{{ a - b - c = 2 }}}
Add (1) and (3)
(1) {{{ a - b - c = 2 }}}
(3) {{{ -a - b + 7c = 8 }}}
{{{ -2b + 6c = 10 }}}
{{{ -b + 3c = 5 }}}
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(2) {{{ -2a + 6b - 2c = 8 }}}
(2) {{{ -a + 3b - c = 4 }}}
I know that {{{ a + b + c = 24 }}}, so add this to (2)
(2) {{{ -a + 3b - c = 4 }}}
{{{ a + b + c = 24 }}}
{{{ 4b = 28 }}}
{{{ b = 7 }}}
and
{{{ -b + 3c = 5 }}}
{{{ -7 + 3c = 5 }}}
{{{ 3c = 12 }}}
{{{ c = 4 }}}
and
(1) {{{ a - b - c = 2 }}}
(1) {{{ a - 7 - 4 = 2 }}}
(1) {{{ a = 13 }}}
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Ann started with $13
Betty started with $7
Carol started with $4
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check:
{{{ a + b + c = 24 }}}
{{{ 13 + 7 + 4 = 24 }}}
{{{ 24 = 24 }}}
OK
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Ann doubled Betty's and Carol's money:
Ann has {{{ 13 - 7 - 4 }}} = $2
Betty has $14
Carol has $8
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Betty doubles carol and Ann's money
Ann has $4
Betty has {{{ 14 - 2 - 8 }}} = $4
Carol has $16
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Carol doubles Ann's and Betty's money
Ann has $8
Betty has $8
Carol has {{{ 16 - 4 - 4 }}} = $8
OK