A number when divided by the sum of $555$ and $445$ gives two times their difference as quotient and $30$ as the remainder. The number is

### Concept & Formula This problem tests your ability to translate compound word statements into specific mathematical variables for the Division Algorithm. $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution * **Given Context translated to variables:** * Divisor = Sum of $555$ and $445$ * Quotient = $2 \times$ (Difference of $555$ and $445$) * Remainder = $30$ * **Calculation:** * Find the Divisor: * Divisor = $555 + 445 = 1000$ * Find the Quotient: * Difference = $555 - 445 = 110$ * Quotient = $2 \times 110 = 220$ * Calculate the final Dividend (The Number): * $\text{Dividend} = (1000 \times 220) + 30$ * $\text{Dividend} = 220000 + 30 = 220030$ ### Exam Strategy & Shortcut Observation is key here. Multiplying anything by $1000$ (the sum) simply shifts the number by three place values (adding three zeros). Knowing the quotient is $220$, the product is instantly $220000$. Adding $30$ only changes the last two digits. You can select $220030$ purely by visual inspection without doing scratchpad arithmetic. ### Common Pitfall A major trap here is misinterpreting "two times their difference". Students might calculate the difference ($110$) and use that directly as the quotient, forgetting to multiply by $2$. Read quantitative phrasing deliberately. ### Final Answer Therefore, the correct answer is **220030**.