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 $(2)$

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

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