Question 1029239
There are twice as many quarters as dollars
Let {{{ d }}} = number of dollars
{{{ d + 2d = 18 }}}
{{{ 3d = 18 }}}
{{{ d = 6 }}}
There are {{{ 18 - 6 = 12 }}} quarters
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The only way to spend and get the ratio
{{{ d/q = 1/2 }}} down to {{{ d/q = 1/3 }}}
is to spend dollars or both dollars and quarters
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I can say:
{{{ ( 6 - x ) / 12 = 1/3 }}}
Multiply both sides by {{{ 12 }}}
{{{ 6 - x = 4 }}}
{{{ x = 2 }}}
Spend just 2 dollars
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Now I can say:
{{{ ( 6 - x ) / ( 12 - y ) = 1/3 }}}
{{{ 3*( 6 - x ) = 12 - y }}}
{{{ 18 - 3x = 12 - y }}}
{{{ 3x - y = 6 }}}
Any whole number solution is good as
long as {{{ x < 6 }}} and {{{ y < 12 }}}
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{{{ x = 4 }}} and
{{{ y = 6 }}} works
spend 4 dollars and 6 quarters
d/q = 2/6
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{{{ x = 5 }}} and
{{{ y = 9 }}} works
spend 5 dollars and 9 quarters
d/q = 1/3
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{{{ x = 3 }}} and
{{{ y = 3 }}} works
spend 3 dollars and 3 quarters
d/q = 3/9
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I think that's it