A good problem to do mentally, without paper...





Since the ages are in the ratio 4/1 to start, and 12 years later 2/1, 12 years MUST represent twice Katie's age (4-2 = 2).   Thus, 12/2 = 6 is Katie's age and Emily is currently 4*6 = 24.



Check:

In 12 years,  Emily will be 24+12 = 36 and Katie will be 6+12 = 18.  

36/18 = 2/1





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Alternate (algebraic method):



     x = Katie's age

    4x = Emily's age



     4x + 12 = 2(x + 12)

     4x + 12 = 2x + 24

          2x = 12  -->  x = 6, 4x = 24,   as above.



 

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