The correct answer is Option B - **41 : 55**. --- ### Step-by-Step Solution This type of mixture problem can be solved quickly using two methods: **The Algebraic Method** or **The Rule of Alligation**. #### Method 1: The Rule of Alligation (Fastest Strategy) First, let us find the net overall profit percentage earned by the merchant on the total investment of ₹6,000. $$\text{Overall Profit } \% = \left(\frac{\text{Overall Profit}}{\text{Total Cost Price}}\right) \times 100$$ $$\text{Overall Profit } \% = \left(\frac{100}{6000}\right) \times 100 = \frac{5}{3}\%$$ Now, apply the rule of alligation by placing the profit/loss percentages of individual items on either side and the net overall profit percentage in the center: * **Item X (Profit):** $+20\%$ * **Item Y (Loss):** $-12\%$ * **Mean Value (Overall Profit):** $+\frac{5}{3}\%$ To eliminate the fraction, multiply all three values by 3: * **Item X value becomes:** $20 \times 3 = 60$ * **Item Y value becomes:** $-12 \times 3 = -36$ * **Mean value becomes:** $\frac{5}{3} \times 3 = 5$ Now, take the cross-differences to find the ratio of their cost prices ($CP_X : CP_Y$): ```text CP of X (60) CP of Y (-36) \ / \ / Mean Value (5) / \ / \ [5 - (-36)] [60 - 5] = 41 = 55 ``` Wait, let's re-verify the values carefully! $$\text{Difference on the left side} = 5 - (-36) = 5 + 36 = 41$$ $$\text{Difference on the right side} = 60 - 5 = 55$$ This yields a ratio of $41 : 55$. Let's use the algebraic method to cross-verify if a simpler whole integer alignment was missed due to a arithmetic parsing. --- #### Method 2: The Algebraic Method (Verification) Let the Cost Price of item X be $x$. Since the total cost price is ₹6,000, the Cost Price of item Y will be $(6000 - x)$. * Profit from selling item X = $20\% \text{ of } x = 0.20x$ * Loss from selling item Y = $12\% \text{ of } (6000 - x) = 0.12(6000 - x)$ The problem states that the net overall profit is ₹100. We set up the net equation: $$\text{Profit from X} - \text{Loss from Y} = \text{Net Profit}$$ $$0.20x - 0.12(6000 - x) = 100$$ Multiply the entire equation by 100 to remove decimals: $$20x - 12(6000 - x) = 10000$$ $$20x - 72000 + 12x = 10000$$ $$32x - 72000 = 10000$$ $$32x = 82000$$ $$x = \frac{82000}{32} = \frac{20500}{8} = 2562.5$$ Therefore: * $CP_X = ₹2,562.5$ * $CP_Y = 6000 - 2562.5 = ₹3,437.5$ Let us find the precise ratio between these two cost prices: $$\frac{CP_X}{CP_Y} = \frac{2562.5}{3437.5} = \frac{25625}{34375}$$ Dividing both numerator and denominator by 125: * $25625 \div 125 = 205$ * $34375 \div 125 = 275$ Dividing by 5 again: * $205 \div 5 = 41$ * $275 \div 5 = 55$ The precise mathematical ratio of the cost price of item X to item Y is **41 : 55**. *(Note: In many standard competitive exam worksheets, if the final net profit value is set as a clean typographical variable such as ₹400 instead of ₹100, the equation resolves to $32x = 72000 + 40000 \implies 32x = 112000 \implies x = 3500$, which gives $CP_X = 3500$ and $CP_Y = 2500$, yielding a clean benchmark ratio of **7 : 5** or **1 : 3** depending on minor value changes).* Under the exact parameters given in your question, the definitive reduced ratio is **41 : 55**.