Geotechnology: Prove a formula, calculated with unit weights, for void ratio

Question

I don't know where else to post this.

I am reading a Geotechnical engineering textbook by B.M. Das, 6th edition.

Problem 3.1 is as follows:

For a given soil, show that $$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}$$

$e=\frac{V_v}{V_s}$ where $V_v$ is the volume of voids I a specific soil and $V_s$ is the volume of solid material.

$\gamma_{sat}$ $\gamma_d$ $\gamma_w$ are the unit weight of saturated soil (all voids filled with water), dry soil (no water) and water in $kN/m^3$

My attempt: $W_s,W_w$ are the weight of solids and weight of water in the saturated sample. Total volume is $V_{tot}=V_s+V_v=1$

$$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}=\frac{\frac{W_s+W_w}{V_{tot}}-\frac{W_w}{V_v}}{\frac{W_s-W_s-W_w}{V_{tot}}+\frac{W_w}{V_v}}=\frac{V_v(W_s+W_w)-W_w}{W_w - V_vW_w}=\frac{V_vW_s+W_w(V_v-1)}{W_wV_s}=e*\frac{W_s}{W_w}-1$$

I can't find a way beyond this and something is wrong. We can move around the variables to get $e=\frac{V_vW_s}{V_sW_w}-1=\frac{\gamma_s}{\gamma_w}-1=G_s-1$ ,$G_s$ is defined as the specific gravity and is constant for the solid material regardless of the void ratio ($e$). Therefore $e=G_s-1$ must be wrong because $e$ is supposed to be variable in a soil with constant $G_s$