Answer :

Let the numbers be x and 5x (as one number is 5 times the other number)
Given, 
If 21 is added to both the numbers, numbers become (x + 21) and (5 x + 21), then one of the new numbers becomes twice the other new number

Then,

21 + 5x = 2(x + 21)
21 + 5x = 2x + 42
5x - 2x = 42 - 21
3x = 21
x = 7

Also, 

5 x = 5(7) = 35

Thus, one number is 7 and the other number is 35.

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