A container is filled with three fluids, X, Y, and Z, mixed in the ratio 8:6:4. From this container, 30 liters of the blend are drawn off. Subsequently, 12 liters of fluid X and 8 liters of fluid Z are introduced into the container. As a result, the volume of fluid X in the final blend exceeds that of fluid Y by 20 liters. Determine the original total volume of the mixture contained in the vessel.

The correct answer is 102L.

Step 1: Understand the initial ratio and fractions of each fluid
The container initially holds three fluids, X, Y, and Z, in the ratio 8 : 6 : 4.
Sum of the ratio terms = 8 + 6 + 4 = 18 parts.

Let $V$ be the original total volume of the mixture in liters.

The proportion of each fluid in the mixture is:

$\text{Fraction of fluid X}=\frac{8}{18}=\frac{4}{9}$

$\text{Fraction of fluid Y}=\frac{6}{18}=\frac{1}{3}$

$\text{Fraction of fluid Z}=\frac{4}{18}=\frac{2}{9}$

Step 2: Calculate the amounts of fluids removed
When 30 liters of the blend are drawn off, the fluids are removed in their respective ratio proportions:

$\text{Amount of X removed}=30\times \frac{8}{18}=\frac{40}{3}\text{ liters}$

$\text{Amount of Y removed}=30\times \frac{6}{18}=10\text{ liters}$

Step 3: Determine the final volumes of fluid X and fluid Y
After removing 30 liters of blend and then adding 12 liters of fluid X:

$\text{Final volume of X}=\left(\frac{4}{9}V-\frac{40}{3}\right)+12=\frac{4}{9}V-\frac{4}{3}$

Since no additional fluid Y was added, the final volume of Y remains:

$\text{Final volume of Y}=\frac{1}{3}V-10$

Step 4: Formulate and solve the equation
It is given that the final volume of fluid X exceeds that of fluid Y by 20 liters:

$\text{Final X}-\text{Final Y}=20$

$\left(\frac{4}{9}V-\frac{4}{3}\right)-\left(\frac{1}{3}V-10\right)=20$

Combine like terms:

$\left(\frac{4}{9}V-\frac{3}{9}V\right)-\frac{4}{3}+10=20$

$\frac{1}{9}V+\frac{26}{3}=20$

$\frac{1}{9}V=20-\frac{26}{3}$

$\frac{1}{9}V=\frac{60}{3}-\frac{26}{3}=\frac{34}{3}$

Solve for $V$:

$V=9\times \frac{34}{3}=3\times 34=102$

Therefore, the original total volume of the mixture contained in the vessel was 102L.