A person bought $80\text{m}$ cloth at $\text{₹}110$ per $\text{m}$ and sold $75\%$ of it at a gain of $6\%$. At what gain percent did he sell the remaining cloth, if he gained $10\%$ on his total outlay?

Let's solve this using the weighted average method. The cost price per meter is constant, so we can work directly with the quantities/proportions of cloth:Part 1 ($75\%$ of cloth) = $\frac{3}{4}$ of totalPart 2 (Remaining $25\%$ of cloth) = $\frac{1}{4}$ of totalThe ratio of the weights is $3:1$.Let the gain percentage on the remaining part be $x\%$.Gain on first part = $6\%$ Overall gain = $10\%$ Using the formula for weighted average:
$$\text{Overall Gain}=\frac{w_1\cdot g_1+w_2\cdot g_2}{w_1+w_2}$$
$$10=\frac{3\times 6+1\times x}{3+1}$$
$$10=\frac{18+x}{4}$$
$$40=18+x$$
$$x=40-18=22\%$$