A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

Let the smaller number be $x$ and the larger number be $5x$.

After adding 21 to both numbers, we have

New smaller number = $(x + 21)$ and new larger number = $(5x + 21)$.

According to problem, we have

$$5x + 21 = 2(x + 21)$$
$$5x + 21 = 2x + 42$$
$$\implies 3x = 21$$
$$\implies x = 7$$

Hence, the numbers are 7 and 35.