In a division problem, the divisor is $7$ times of quotient and $5$ times of remainder. If the dividend is $6$ times of remainder, then the quotient is equal to

### Concept & Logic This problem is solved by translating word statements into algebraic equations and substituting them into the standard Division Algorithm. $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution * **Given Relationships:** * Divisor ($D$) = $5 \times$ Remainder ($R$) * Divisor ($D$) = $7 \times$ Quotient ($Q$) * Dividend = $6 \times$ Remainder ($R$) * **Calculation:** * Substitute the variables expressed in terms of $R$ into the division formula: * $6R = (5R \times Q) + R$ * Subtract $R$ from both sides of the equation: * $6R - R = 5RQ$ * $5R = 5RQ$ * Assuming $R$ is not zero (as that would mean the divisor is zero, which is undefined), we can divide both sides by $5R$: * $Q = 1$ ### Exam Strategy & Shortcut Algebraic substitution is the fastest path here. By recognizing that both the Dividend and Divisor can be expressed in terms of the Remainder ($R$), you can quickly set up the equation $6R = 5RQ + R$. The $R$ terms cancel out almost instantly in your head, leaving $5 = 5Q$, which immediately gives $Q = 1$. ### Common Pitfall A common trap is attempting to equate $7Q = 5R$ and solving for a numerical value prematurely without using the Dividend information. Always write down all given conditions before attempting to solve the system of equations. ### Final Answer Therefore, the correct answer is **1**.