If $6*43 - 46@9 = 1904$, which of the following should come in place of $*$?

### Concept & Logic This is a cryptarithm involving missing digits in subtraction. The most systematic approach is to rewrite the numbers using their place values, isolating the unknown digits ($*$ and $@$) into a simple linear algebraic equation. ### Step-by-Step Solution * **Place Value Expansion:** Expand the terms into thousands, hundreds, tens, and units. * $6*43 = 6000 + 100(*) + 40 + 3 = 6043 + 100(*)$ * $46@9 = 4000 + 600 + 10(@) + 9 = 4609 + 10(@)$ * **Form the Equation:** Substitute these expansions back into the original equation. $$(6043 + 100(*)) - (4609 + 10(@)) = 1904$$ * **Group Constants:** $$(6043 - 4609) + 100(*) - 10(@) = 1904$$ $$1434 + 100(*) - 10(@) = 1904$$ * **Isolate Variables:** $$100(*) - 10(@) = 1904 - 1434$$ $$100(*) - 10(@) = 470$$ * **Simplify:** Divide the entire equation by 10. $$10(*) - @ = 47$$ * **Deduce Digits:** Since $*$ and $@$ are single digits ($0$ through $9$): * If $* = 4$, then $40 - @ = 47 \implies @ = -7$ (Invalid, must be positive). * If $* = 5$, then $50 - @ = 47 \implies @ = 3$ (Valid, both are single digits). * If $* = 6$, then $60 - @ = 47 \implies @ = 13$ (Invalid, must be $<10$). * **Conclusion:** The digit $*$ must be $5$. Looking at our options ($4, 6, 9$, Cannot be determined), $5$ is missing. ### Exam Strategy & Shortcut **Column Subtraction Analysis:** Look directly at the hundreds column of the subtraction. The given equation is roughly $6000 - 4600 = 1900$. Let's align it vertically: $6 * 4 3$ $- 4 6 @ 9$ $-------$ $1 9 0 4$ Notice the thousands column: $6 - 4$ is normally $2$, but the result is $1$. This means a carry (borrow) was taken from the $6$ for the hundreds column. For the hundreds column: We borrowed $1$, so it becomes $(10 + *) - 6$. Wait, did the tens column borrow from the hundreds? Tens: $4 - @ = 0$ (after borrowing $1$ for the units $13-9=4$). So $3 - @$ needs to yield $0$, or it borrows. Since $10(*) - @ = 47$ derived earlier is foolproof, relying on pure algebra here is actually safer than mental column tracking. ### Common Pitfall Choosing "Cannot be determined" simply because there are two variables in one equation. Because the variables are constrained strictly to single positive digits ($0-9$), Diophantine equations like $10x - y = 47$ often have only one unique valid solution! ### Final Answer Therefore, the correct answer is **None of these**.