Fast multiplication using 10,000 − 1 trick: Evaluate 469157 × 9999 accurately without long multiplication.

Introduction / Context: Products involving 9999 suggest a classic shortcut: 9999 = 10000 − 1. This allows a quick calculation using place-value shifting and a single subtraction instead of full long multiplication.

Given Data / Assumptions:

• a = 469157
• Compute a * 9999.
• Use the identity 9999 = 10000 − 1.

Concept / Approach: Rewrite a * 9999 as a * 10000 − a. Multiplication by 10000 shifts the number four places to the left (because 10000 = 10^4). Then subtract the original number once. This avoids cumbersome partial products and reduces error probability.

Step-by-Step Solution:Compute a * 10000 = 469157 * 10000 = 4,691,570,000.Subtract a: 4,691,570,000 − 469,157 = 4,691,100,843.Match with options: 4,691,100,843 corresponds to 4691100843.

Verification / Alternative check: Reverse check by adding back: 4,691,100,843 + 469,157 = 4,691,570,000, which equals 469157 * 10000. Therefore the transformation using 10000 − 1 is validated.

Why Other Options Are Wrong:Values close in magnitude but with swapped inner digits indicate common subtraction misalignments.Any option not ending with 0843 is suspect because subtracting 469,157 from an exact hundred-million multiple naturally yields the final four digits 0843.

Common Pitfalls: Misaligning place values during subtraction; forgetting that multiplying by 10000 appends four zeros; performing a − b as b − a by mistake.

Final Answer: 4691100843