$414 \times x \times 7 = 127512$

### Concept & Strategy This algebraic equation requires isolating the unknown variable $x$. However, completely resolving the multiplication and subsequent division is highly time-consuming. The fastest approach is utilizing the unit digit methodology. $$ A \times x \times B = C $$ ### Step-by-Step Solution * **Given:** The equation $414 \times x \times 7 = 127512$. * **Step 1:** Multiply the known terms on the left side. $414 \times 7 = 2898$ * **Step 2:** Substitute back into the equation. $2898 \times x = 127512$ * **Step 3:** Isolate $x$ by dividing both sides by $2898$. $x = \frac{127512}{2898}$ * **Step 4:** Perform the division. $x = 44$ ### Exam Strategy & Shortcut Use **Unit Digit Analysis** to bypass the long division entirely. Look at the given equation: $414 \times x \times 7 = 127512$. 1. Multiply the unit digits of the known numbers on the left: $4 \times 7 = 28$. The left side multiplier has a unit digit of $8$. 2. The right side of the equation ends in $2$ ($127512$). 3. Therefore, $8 \times (\text{unit digit of } x)$ must result in a number ending in $2$. 4. Check multiplication tables for $8$: $8 \times 4 = 32$ and $8 \times 9 = 72$. So $x$ must end in $4$ or $9$. 5. Look at the options: (a) 36, (b) 40, (c) 44, (d) 48. Only option (c) ends in $4$. You can mark it immediately. ### Common Pitfall The most common mistake is performing the full long division ($127512 \div 2898$) under time pressure. This not only wastes valuable minutes but also heavily increases the probability of a calculation error. ### Final Answer Therefore, the correct answer is **44**.