Question 133068
Katie's age now: k
Anne's age now: a


Katie's age 4 years ago: k - 4
Anne's age 4 years ago: a - 4


Anne's age 6 years from now: a + 6


Four years ago, katie was twice as old as Anne was then:  {{{k-4=2(a-4)}}}
In 6 years Anne will be the same age as Katie is now: {{{k=a+6}}}


{{{k-4=2(a-4)}}}
{{{k-4=2a-8}}}
{{{k-2a=-4}}}


Since we know that  {{{k=a+6}}}, we can substitute:
{{{cartoon(red(k)-2a=-4,red((a+6))-2a=-4)}}}


Collect like terms:
{{{a-2a=-4-6}}}
{{{-a=-10}}}


Multiply both sides by -1:
{{{a=10}}}, and {{{k=a+6=10+6=16}}}


So Katie is 16 and Anne is 10.


Check:
Four years ago, katie was twice as old as Anne was then:


16 - 4 = 12 
2 * (10 - 4) = 2 * 6 = 12


In 6 years Anne will be the same age as Katie is now:


10 + 6 = 16


Answer checks.