Swati and Rajni enter into a partnership with their capitals in the ratio 5 : 6. At the end of 7 months Swati withdraws her capital. If they receive the profit in the ratio of 5 : 9, find how long was Rajni's capital used.

### Concept & Profit Ratio Formula The ratio of profits is equal to the ratio of the products of capital and the time it was invested. $$\frac{Profit_A}{Profit_B} = \frac{Capital_A \times Time_A}{Capital_B \times Time_B}$$ ### Step-by-Step Solution 1. The ratio of initial capitals is $C_{Swati} : C_{Rajni} = 5 : 6$. Let Swati's capital be 5 units and Rajni's capital be 6 units. 2. Swati invested for 7 months, so $Time_{Swati} = 7$. 3. Let Rajni's money be used for $t$ months, so $Time_{Rajni} = t$. 4. The profit ratio is given as $Profit_{Swati} : Profit_{Rajni} = 5 : 9$. 5. Substituting the values into the formula: $\frac{5 \times 7}{6 \times t} = \frac{5}{9}$ 6. Simplify by cancelling the 5 in the numerators: $\frac{7}{6t} = \frac{1}{9}$ 7. Cross-multiply: $6t = 63$ 8. $t = \frac{63}{6} = 10.5$ months. 9. Since 10.5 months is not among the given numerical options (10, 12, 14), the answer is "None of these". ### Exam Strategy & Shortcut Write down the relationship directly: $(5 \times 7) / (6 \times t) = 5 / 9$. Mentally drop the 5s. You have $7 / 6t = 1 / 9 \Rightarrow 6t = 63 \Rightarrow t = 10.5$. It takes less than 15 seconds to realize the answer isn't an integer, pointing straight to "None of these". ### Common Pitfall Guessing "12 months" or "14 months" simply because $10.5$ is a fraction, assuming a calculation error was made, instead of confidently trusting the math and selecting "None of these". ### Final Answer Therefore, the correct answer is **None of these**.