The ratio of two numbers is $1:2$. If $21$ is added to both numbers, then the ratio becomes $5:7$. Find these numbers.

Let the two numbers be $x$ and $2x$ based on the initial ratio $1:2$.
According to the question, if $21$ is added to both numbers, the new ratio becomes $5:7$:

$$\frac{x+21}{2x+21}=\frac{5}{7}$$

Cross-multiply to solve for $x$:

$$7(x+21)=5(2x+21)$$

$$7x+147=10x+105$$

$$147-105=10x-7x$$

$$42=3x$$

$$x=14$$

Now substitute $x$ back to get the numbers:

$$\text{First number}=x=14$$

$$\text{Second number}=2x=2\times 14=28$$

Therefore, the numbers are $14$ and $28$.