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 quantities in Container X in terms of a variable.

The proportion of juice to water in container X is given as 9 : 5.

Let the volume of juice in container X be:

${\mathrm{Juice}}_X=9k$

Let the volume of water in container X be:

${\mathrm{Water}}_X=5k$

Therefore, the total volume of liquid in container X is:

${\mathrm{Total}}_X=9k+5k=14k$

Step 2: Determine the volume of juice and water in Container Y.

Container Y contains 20% more juice than container X:

${\mathrm{Juice}}_Y=9k\times \left(1+0.20\right)=9k\times 1.2=10.8k$

Container X contains 25% more water than container Y, which means:

${\mathrm{Water}}_X={\mathrm{Water}}_Y\times \left(1+0.25\right)=1.25\times {\mathrm{Water}}_Y$

Substitute ${\mathrm{Water}}_X=5k$ to find the water in container Y:

${\mathrm{Water}}_Y=\frac{5k}{1.25}=4k$

Step 3: Solve for variable k using the combined volume of water.

The combined volume of water in both containers is 54 liters:

${\mathrm{Water}}_X+{\mathrm{Water}}_Y=54$

$5k+4k=54$

$9k=54$

$k=6$

Step 4: Calculate the total volume of liquid in Container X.

Substitute $k=6$ into the total volume expression for container X:

${\mathrm{Total}}_X=14k=14\times 6=84\text{liters}$