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

Options

A. `(11d)/(7)`
B. `(18d)/(11)`
C. `(13d)/(9)`
D. `(23d)/(18)`

Answer

To find the relative density of a mixture of three liquids with densities \(d\), \(2d\), and \(3d\) mixed in equal proportions by weight, we can follow these steps:

### Step 1: Define the Mass of Each Liquid
Assume we take an equal mass \(m\) of each liquid. Therefore:
- Mass of liquid 1 (density \(d\)) = \(m\)
- Mass of liquid 2 (density \(2d\)) = \(m\)
- Mass of liquid 3 (density \(3d\)) = \(m\)

### Step 2: Calculate the Volume of Each Liquid
Using the formula for volume \(V = \frac{m}{\text{density}}\):
- Volume of liquid 1, \(V_1 = \frac{m}{d}\)
- Volume of liquid 2, \(V_2 = \frac{m}{2d}\)
- Volume of liquid 3, \(V_3 = \frac{m}{3d}\)

### Step 3: Calculate the Total Volume of the Mixture
The total volume \(V_{total}\) of the mixture is the sum of the volumes of the three liquids:
\[ V_{total} = V_1 + V_2 + V_3 = \frac{m}{d} + \frac{m}{2d} + \frac{m}{3d} \]

### Step 4: Find a Common Denominator
To add these fractions, we find a common denominator, which is \(6d\):
\[ V_{total} = \frac{6m}{6d} + \frac{3m}{6d} + \frac{2m}{6d} = \frac{(6 + 3 + 2)m}{6d} = \frac{11m}{6d} \]

### Step 5: Calculate the Total Mass of the Mixture
The total mass \(M_{total}\) of the mixture is simply the sum of the masses of the three liquids:
\[ M_{total} = m + m + m = 3m \]

### Step 6: Calculate the Relative Density of the Mixture
The relative density \(D_{relative}\) of the mixture is given by the formula:
\[ D_{relative} = \frac{M_{total}}{V_{total}} = \frac{3m}{\frac{11m}{6d}} = \frac{3m \cdot 6d}{11m} \]
Here, \(m\) cancels out:
\[ D_{relative} = \frac{18d}{11} \]

### Conclusion
Thus, the relative density of the mixture is \(\frac{18d}{11}\).

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