mass of liquid with density \(\beta= M _2\)

Total volume \(= V\)

Net density of mixture \(=\sigma\)

Total mass \(= M _1+ M _2\)

\(\Rightarrow V \sigma= M _1+ M _2\)

\(\Rightarrow M _2= V \sigma- M _1 \ldots \ldots(1)\)

\(\left[\because \frac{\text { Total Mass }}{ v }=\sigma\right]\)

\(T=\frac{T \text { otal mass }}{\text { Total volume }}=\frac{M_1+M_2}{\frac{M_1}{\alpha}+\frac{M_2}{\beta}} \ldots \ldots \text { (2) }\)

sub \((1)\) in \((2)\)

\(\Rightarrow \sigma=\frac{ M _1+\left( v \sigma- M _1\right)}{\frac{ M _1}{\alpha}+\left(\frac{ v \sigma- M _1}{\beta}\right)}\)

\(\Rightarrow M _1=\frac{\alpha V (\beta-\sigma)}{\beta-\alpha} \text {. }\)