Three liquids of densities d, 2d and 3d are mixed in equal proportions of weights. The relative density of the mixture is

The correct option is 18d/11.

Let us determine the density of the mixture step-by-step.

Step 1: Understand the given values
We have three liquids mixed in equal proportions of weights (masses). Let the mass of each liquid be $m$.
The densities of the three liquids are:
First liquid density, ${\rho }_1=d$
Second liquid density, ${\rho }_2=2d$
Third liquid density, ${\rho }_3=3d$

Step 2: Find the volumes of each liquid
Since volume is defined as mass divided by density ($V=\frac{m}{\rho }$), the volumes of the three liquids are:

$V_1=\frac{m}{d}$

$V_2=\frac{m}{2d}$

$V_3=\frac{m}{3d}$

Step 3: Calculate the total mass and total volume of the mixture
The total mass of the mixture is:

$M_{\text{total}}=m+m+m=3m$

The total volume of the mixture is:

$V_{\text{total}}=V_1+V_2+V_3$

$V_{\text{total}}=\frac{m}{d}+\frac{m}{2d}+\frac{m}{3d}$

Taking $\frac{m}{d}$ as a common factor:

$V_{\text{total}}=\frac{m}{d}\left(1+\frac{1}{2}+\frac{1}{3}\right)$

To add the fractions inside the parentheses, find a common denominator (which is 6):

$V_{\text{total}}=\frac{m}{d}\left(\frac{6+3+2}{6}\right)=\frac{11m}{6d}$

Step 4: Calculate the density of the mixture
The density of the mixture is the total mass divided by the total volume:

${\rho }_{\text{mixture}}=\frac{M_{\text{total}}}{V_{\text{total}}}$

${\rho }_{\text{mixture}}=\frac{3m}{\left(\frac{11m}{6d}\right)}$

Simplifying the expression by cancelling the common term $m$ and rearranging:

${\rho }_{\text{mixture}}=3\times \frac{6d}{11}=\frac{18d}{11}$