Which of the following digits will replace the $H$ marks in the following equation? $$9H + H8 + H6 = 230$$

### Concept & Logic This is a cryptarithm or number puzzle where a letter represents a specific single digit ($0$ through $9$). The most robust way to solve this is to break down each two-digit number into its expanded place-value form (Tens and Units). A two-digit number like "$XY$" can be written algebraically as $10X + Y$. ### Step-by-Step Solution * **Given Equation:** $9H + H8 + H6 = 230$ * **Place Value Expansion:** Expand each term based on its digit positions. * $9H = (9 \times 10) + H = 90 + H$ * $H8 = (H \times 10) + 8 = 10H + 8$ * $H6 = (H \times 10) + 6 = 10H + 6$ * **Form the Algebraic Equation:** Substitute the expanded forms back into the original sum. $$(90 + H) + (10H + 8) + (10H + 6) = 230$$ * **Group Like Terms:** $$21H + 104 = 230$$ * **Solve for H:** $$21H = 230 - 104$$ $$21H = 126$$ $$H = \frac{126}{21} = 6$$ * **Conclusion:** The digit that replaces $H$ is $6$. Looking at our given options (3, 4, 5, 9), $6$ is not listed. ### Exam Strategy & Shortcut **Unit Digit Method:** Instead of full algebra, just look at the unit digits of the numbers being added! The unit digits are: $H, 8,$ and $6$. Their sum must result in a number ending in the unit digit of the total, which is $0$. So, $H + 8 + 6 = H + 14$. For $H + 14$ to end in a $0$, $H$ must be $6$ (since $6 + 14 = 20$). Since $6$ is not in the options, the answer is immediately "None of these". This takes 5 seconds! ### Common Pitfall A common mistake is trying to randomly substitute the options into the equation one by one. While this eventually works, it is time-consuming. Using the unit digit trick bypasses all trial and error instantly. ### Final Answer Therefore, the correct answer is **None of these**.