Next 2 years,the sum of Madison's age and 3 times Harper's age is:

       47+2+2 x 3=55(ages)

Put the  Madison's ages is :a

Put the Harper's ages is:b

We have:a+3b=55\(\Rightarrow\)5b=55\(\Rightarrow\)b=11

So the Harper's age is 11 ages

Tell the age of Madison and Harper sequence is a and b

\(\Rightarrow\left\{{}\begin{matrix}a+3a=47\\\dfrac{a+2}{b+2}=2\end{matrix}\right.\)

=> a + 2 = 2b + 4

=> a = 2b + 2

=> a + 3b = 2b + 2 + 3b = 47

=> 5b = 45 

=> b = 9

=> a = 47 - 3.9 = 20

So the age of Harper is 9