A chemical storage vessel contains three industrial solvents—Solvent A, Solvent B, and Sol… — GMAT

A chemical storage vessel contains three industrial solvents—Solvent A, Solvent B, and Solvent C—blended in the initial ratio of $2 : 5 : 7$ , respectively. After $24$ liters of Solvent A and $12$ liters of Solvent B are added to the vessel, the ratio of Solvent A to Solvent B becomes $4 : 7$ . What was the initial volume, in liters, of Solvent C in the vessel?

Answer: 140 liters

Answer

The initial volume of Solvent C in the vessel was 140 liters.

Defining the initial amounts of Solvents A, B, and C as

2x

5x

, and

7x

, respectively, the addition of

24

liters of Solvent A and

12

liters of Solvent B yields the ratio equation

\frac{2x + 24}{5x + 12} = \frac{4}{7}

. Cross-multiplying gives

14x + 168 = 20x + 48

, which simplifies to

6x = 120

and

x = 20

. The initial volume of Solvent C is therefore

7 \times 20 = 140

liters.

Step-by-Step Solution

Express the initial quantities of the three solvents in terms of a common ratio multiplier

x

Solvent A =

2x

, Solvent B =

5x

, Solvent C =

7x

A ratio of

2 : 5 : 7

implies the actual quantities are proportional to 2, 5, and 7 by a factor

x

Set up an equation representing the modified ratio of Solvent A to Solvent B.

\frac{2x + 24}{5x + 12} = \frac{4}{7}

Adding 24 liters to Solvent A yields

2x + 24

liters, and adding 12 liters to Solvent B yields

5x + 12

liters.

Solve the algebraic equation for

x

7(2x + 24) = 4(5x + 12) \implies 14x + 168 = 20x + 48 \implies 6x = 120 \implies x = 20

Cross-multiplication converts the ratio comparison into a linear equation.

Compute the initial volume of Solvent C.

Initial volume of Solvent C =

7x = 7(20) = 140

liters.

Substitute

x = 20

into the expression

7x

for the original amount of Solvent C.

Key Concept

Ratio scaling and algebraic modeling of partial additions

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