Find the missing number in the following addition problem: $$ \begin{matrix} & 8 & 3 & 5 \\ & 4 & * & 8 \\ + & 9 & * & 4 \\ \hline 2 & 2 & * & 7 \end{matrix} $$

### Concept & Logic This is a cryptarithm addition puzzle. The missing asterisk ($*$) represents the same digit in all three places. We can deduce this digit by analyzing the vertical sums column by column, starting from the units place and tracking the carry-overs. ### Step-by-Step Solution * **Units Column:** Add the visible digits: $5 + 8 + 4 = 17$. The result has a $7$ at the bottom, and $1$ is carried over to the tens column. * **Tens Column:** Form an algebraic equation including the carry-over. $1 \text{ (carry)} + 3 + * + * = \text{number ending in } *$ $4 + 2* = 10 \text{ (carry out)} + *$ Subtract $*$ from both sides: $4 + * = 10 \implies * = 6$ * **Verification (Hundreds Column):** Let us verify if $* = 6$ works for the rest of the problem. If $* = 6$, the tens sum is $1 + 3 + 6 + 6 = 16$. The bottom digit is $6$ (matching our $*$), and $1$ is carried over. Hundreds sum: $1 \text{ (carry)} + 8 + 4 + 9 = 22$. This perfectly matches the $22$ at the bottom left of the result. ### Exam Strategy & Shortcut **Option Elimination:** Instead of forming equations, plug the given options into the tens column logic: $1 (\text{carry}) + 3 + 2 \times (\text{option}) = \text{number ending in that option}$. * Try (a) 0: $4 + 2(0) = 4$. (Ends in 4, not 0) * Try (b) 4: $4 + 2(4) = 12$. (Ends in 2, not 4) * Try (c) 6: $4 + 2(6) = 16$. (Ends in 6! Match found instantly). ### Common Pitfall Forgetting the carry-over from the units column ($1$) when summing the tens column is the most frequent error. This leads to an incorrect equation like $3 + 2x = 10k + x$, which throws off the entire calculation. ### Final Answer Therefore, the correct answer is **6**.