In a division sum, the divisor is ten times the quotient and five times the remainder. If the remainder is $46$, determine the dividend.

### Concept & Formula The fundamental division algorithm states that the original number (dividend) is the product of the divisor and quotient, plus any remainder: $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution **Given:** * $\text{Remainder} = 46$ * $\text{Divisor} = 10 \times \text{Quotient}$ * $\text{Divisor} = 5 \times \text{Remainder}$ **Calculation / Deduction:** * First, calculate the exact value of the Divisor: $$ \text{Divisor} = 5 \times 46 = 230 $$ * Next, use the relationship between the Divisor and the Quotient to find the Quotient: $$ 230 = 10 \times \text{Quotient} $$ $$ \text{Quotient} = \frac{230}{10} = 23 $$ * Finally, substitute all three known values into the core division formula: $$ \text{Dividend} = (230 \times 23) + 46 $$ $$ \text{Dividend} = 5290 + 46 = 5336 $$ ### Exam Strategy & Shortcut To skip the final large multiplication ($230 \times 23$), use the unit digit method. The unit digit of the divisor is $0$, the quotient is $3$, and the remainder is $6$. The unit digit of the dividend must be $(0 \times 3) + 6 = 6$. If $5336$ is the only option ending in $6$, you can confidently select it without doing the full addition. ### Common Pitfall A common error is mapping the relationships backward, such as assuming the "quotient is ten times the divisor," which completely corrupts the calculation chain. Always write out the equations exactly as the sentence is structured. ### Final Answer Therefore, the correct answer is 5336.