$19.99 \times 9.9 + 99.9 = $x$

Concept & Logic When dealing with decimals that are very close to whole numbers (like 19.99 and 9.9), use algebraic expansion to convert the multiplication into subtraction from a round number, making it much easier to calculate. $$(a - x)(b - y) = ab - ay - bx + xy$$ Step-by-Step Solution * Rewrite the difficult decimals as whole numbers minus a small decimal: $$19.99 = (20 - 0.01)$$ $$9.9 = (10 - 0.1)$$ * Substitute these into the multiplication part of the expression: $$(20 - 0.01) \times (10 - 0.1)$$ * Expand using the distributive property (FOIL): $$200 - 2 - 0.1 + 0.001$$ * Simplify the expression: $$197.901$$ * Now add the remaining part of the original equation ($99.9$): $$197.901 + 99.9$$ * Add 100 and subtract 0.1 to make the addition easier: $$197.901 + 100 - 0.1 = 297.901 - 0.1 = 297.801$$ Exam Strategy & Shortcut Use pure approximation to eliminate wrong options instantly. Assume $19.99 \approx 20$ and $9.9 \approx 10$. $20 \times 10 = 200$. Add $99.9 \approx 100$. The answer must be very close to $300$ ($200 + 100 = 300$). Looking at the options, only (b) $297.801$ and (d) $296.910$ are close. Check the unit digit of the decimal multiplication: $9 \times 9 = 81$, meaning the product ends in `1`. The addition of $99.9$ (which is $99.900$) won't change the final decimal digit, so it must end in `1`. Only option (b) ends in `1`. Common Pitfall Attempting long-form multiplication with $19.99 \times 9.9$ wastes valuable time and drastically increases the chance of misplacing the decimal point. Always look to approximate first to narrow down the options. Final Answer **Therefore, the correct answer is 297.801.**