Solve $1576 \div 45.02 + 23.99 \times \sqrt{255} = x$

### Concept & Strategy The problem involves complex decimal numbers and square roots. The key insight is to recognize this as an **Approximation** problem. Use the BODMAS rule alongside rounding numbers to their nearest manageable integers. ### Step-by-Step Solution * **Given Equation:** $$1576 \div 45.02 + 23.99 \times \sqrt{255} = x$$ * **Approximation Step:** Round the numbers to the nearest integers or perfect squares to simplify calculations: * $1576$ can be rounded to $1575$ (since it is easily divisible by $45$). * $45.02$ rounds to $45$. * $23.99$ rounds to $24$. * $\sqrt{255}$ is extremely close to $\sqrt{256}$, which is exactly $16$. * **Calculation:** Substituting the approximate values back into the equation: $$x \approx 1575 \div 45 + 24 \times 16$$ Using BODMAS, perform division and multiplication first: $$1575 \div 45 = 35$$ $$24 \times 16 = 384$$ Add the results together: $$x \approx 35 + 384 = 419$$ * **Final Match:** The calculated value $419$ is closest to option $420$. ### Exam Strategy & Shortcut In competitive exams, decimal values ending in `.99` or `.02` are glaring signals to use approximation. Instead of doing exact division, estimate $1576 \div 45$ by thinking $45 \times 10 = 450$, $45 \times 30 = 1350$, remaining is roughly $225$ which is $5 \times 45$. Total $35$. Calculate $24 \times 16$ mentally as $(20+4)\times 16 = 320 + 64 = 384$. Sum is $419 \approx 420$. ### Common Pitfall Attempting to calculate the exact decimal values will consume a massive amount of time. Another mistake is ignoring the BODMAS rule and adding terms before finishing the division or multiplication. ### Final Answer **Therefore, the correct answer is 420.**