The average of $7$ numbers is $30$. If one number is excluded, the average becomes $31$. What is the excluded number?

1. Sum of all $7$ numbers $=7\times 30=210$.
2. After excluding one number, $6$ numbers remain, and their average is $31$.
3. Sum of the remaining $6$ numbers $=6\times 31=186$.
4. Excluded number $=\text{Sum of 7 numbers}-\text{Sum of 6 numbers}=210-186=24$.