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Let  f  be the age of the father and  s  be the age of the son.

 

The ratio of the age of the father to the age of the son 2 years ago was  10:1

 

The ratio of   f - 2   to   s - 2   =  10:1

 

f - 2 : s - 2   =   10:1

 

(f - 2) / (s - 2)  =  10/1

 

(f - 2) / (s - 2)  =  10

 

f - 2   =   10(s - 2)

 

f - 2   =   10s - 20

 

f   =   10s - 18

 

In 5 years, the age of the father will be  4  times the age of the son.

 

f + 5   =   4(s + 5)

 

f + 5   =   4s + 20

 

f   =   4s + 15

 

Now since   f  =  10s -18   and   f  =  4s + 15   we can say...

 

10s - 18  =  4s + 15

 

10s - 4s  =  15 + 18

 

6s  =  33

 

s  =  5.5

 

Now we can find the value of  f  using this value of  s

 

f   =   4s + 15   =   4(5.5) + 15   =   37

 

the difference in the age of the father and son   =   f - s   =   37 - 5.5   =   31.5

(f-2)  = (10)(s-2)     f= 10s - 18

 

f+5 = 4(s+5)

f= 4s +15        equate the two f definitions     (10s-18) = 4s+15        solve for s = 5.5     so f = 37     Difference = 31.5 years