In container X, the proportion of juice to water is 9 : 5. Container Y contains 20% more juice than container X, while container X contains 25% more water than container Y. If the combined volume of water in both containers is 54 liters, what is the total volume of the liquid in container X?

The correct option is 84 liters.

Step 1: Express the volume of liquids in Container X.
The ratio of juice to water in Container X is given as 9 : 5.
Let the common multiplier be $k$.
Volume of juice in Container X = $9k$
Volume of water in Container X = $5k$
Therefore, the total volume of liquid in Container X is:
$\text{Total volume in Container X}=9k+5k=14k$

Step 2: Express the volume of water in Container Y.
We are given that Container X contains 25% more water than Container Y.
This relationship can be written as:
$\text{Water in Container X}=\text{Water in Container Y}+25\%\text{of Water in Container Y}$
$\text{Water in Container X}=1.25\times \left(\text{Water in Container Y}\right)$
Now, expressing the water in Container Y in terms of $k$:
$\text{Water in Container Y}=\frac{\text{Water in Container X}}{1.25}$
$\text{Water in Container Y}=\frac{5k}{1.25}=4k$

Step 3: Solve for $k$ using the combined water volume.
The combined volume of water in both containers is 54 liters:
$\text{Water in Container X}+\text{Water in Container Y}=54$
Substitute the values in terms of $k$:
$5k+4k=54$
$9k=54$
$k=\frac{54}{9}=6$

Step 4: Calculate the total volume of liquid in Container X.
Now substitute $k=6$ back into the total volume expression for Container X:
$\text{Total volume in Container X}=14k=14\times 6=84\text{liters}$

Thus, the total volume of the liquid in container X is 84 liters.