A chemical storage vessel contains a mixture composed of three liquid components: Componen… — GRE General Test

A chemical storage vessel contains a mixture composed of three liquid components: Component $X$ , Component $Y$ , and Component $Z$ . Initially, Component $X$ accounts for $\frac{3}{8}$ of the total mixture volume, and Component $Y$ accounts for $\frac{5}{12}$ of the total mixture volume, with Component $Z$ occupying the remainder of the volume.

During a processing stage, $\frac{1}{3}$ of Component $X$ is extracted and $\frac{2}{5}$ of Component $Y$ is extracted, while Component $Z$ remains completely unchanged. If the total volume of the mixture remaining in the vessel after processing is 170 milliliters, what was the total volume, in milliliters, of the mixture in the vessel before processing?

Answer: 240 milliliters

Answer

The initial total volume of the mixture was 240 milliliters.

To find the initial total volume, first find the fraction of the initial mixture that is Component Z:

1 - (3/8 + 5/12) = 5/24

. Next, compute the remaining amounts of each component relative to the initial total volume

V

: Component X has

(1 - 1/3) \times 3/8 = 1/4 = 6/24

remaining; Component Y has

(1 - 2/5) \times 5/12 = 1/4 = 6/24

remaining; Component Z retains its

5/24

. Summing these yields

6/24 + 6/24 + 5/24 = 17/24

of the original volume. Setting

(17/24)V = 170

mL gives

V = 170 \times (24/17) = 240

mL.

Step-by-Step Solution

Find the initial fraction of the mixture representing Component

Z

Component

Z = 1 - \left(\frac{3}{8} + \frac{5}{12}\right) = 1 - \left(\frac{9}{24} + \frac{10}{24}\right) = 1 - \frac{19}{24} = \frac{5}{24}

The sum of all component fractions must equal 1.

Calculate the remaining fraction for each component relative to the initial total volume

V

Remaining

X = \left(1 - \frac{1}{3}\right) \times \frac{3}{8}V = \frac{2}{3} \times \frac{3}{8}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Y = \left(1 - \frac{2}{5}\right) \times \frac{5}{12}V = \frac{3}{5} \times \frac{5}{12}V = \frac{1}{4}V = \frac{6}{24}V

.
Remaining

Z = \frac{5}{24}V

When a fraction of a component is removed, the remaining fraction of that component is multiplied by its original portion of the total mixture.

Sum the remaining component fractions to find the total remaining volume as a fraction of

V

Total Remaining Fraction

= \frac{6}{24}V + \frac{6}{24}V + \frac{5}{24}V = \frac{17}{24}V

Combining the remaining amounts gives the fraction of the initial mixture left.

Solve for the initial total volume

V

using the given final volume of 170 mL.

\frac{17}{24}V = 170 \implies V = 170 \times \frac{24}{17} = 10 \times 24 = 240 \text{ mL}.

Multiplying the final volume by the reciprocal of the remaining fraction yields the original volume.

Key Concept

Multi-step fraction operations and solving for original whole amounts

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