A laboratory mixture is prepared by combining three liquid solutions: Solution X, Solution… — ACT

A laboratory mixture is prepared by combining three liquid solutions: Solution X, Solution Y, and Solution Z. Initially, Solution X makes up $20\%$ of the total volume of the mixture. Solution Y's volume is $\frac{1}{3}$ of the combined volume of Solution X and Solution Z, and Solution Z makes up the remaining portion of the mixture. If $3.0$ liters of Solution Z are added to the mixture, Solution Z then constitutes exactly $58\%$ of the new total volume. What was the original total volume, in liters, of the mixture before Solution Z was added?

A
15
B
24
C
35
42Answer
E
100

Answer

The original total volume of the mixture was 42 liters.

The correct option represents the original volume of

42

liters. Setting the original total volume to

V

, we find that Solution X is

0.20V

. Since Solution Y is one-third of the combined volume of X and Z, we write

Y = \frac{1}{3}(V - Y)

, which simplifies to

Y = 0.25V

. The remaining portion, Solution Z, must be

V - 0.20V - 0.25V = 0.55V

. Adding

3.0

liters of Solution Z increases both the volume of Solution Z and the total volume of the mixture, yielding the proportion

(0.55V + 3) / (V + 3) = 0.58

. Solving this linear equation results in

0.03V = 1.26

, which gives

V = 42

Step-by-Step Solution

Express the initial fraction of Solution X as a decimal.

Solution X makes up

0.20

of the total volume.

Converting

20\%

to a decimal simplifies subsequent algebraic modeling.

Set up an equation for the fraction of Solution Y in terms of the total volume

V

Solution Y makes up

0.25

of the total volume.

If Solution Y's volume is

\frac{1}{3}

of the combined volume of Solution X and Solution Z, then

Y = \frac{1}{3}(V - Y)

. Multiplying by

3

gives

3Y = V - Y

, which simplifies to

4Y = V

, or

Y = 0.25V

Determine the initial fraction of Solution Z in the mixture.

Solution Z initially makes up

0.55

of the total volume.

Subtracting the fractions of Solution X and Solution Y from the whole yields

1 - 0.20 - 0.25 = 0.55

Set up and solve the equation representing the addition of Solution Z.

V = 42

liters

Adding

3.0

liters of Solution Z increases the volume of Solution Z to

0.55V + 3

and the total volume to

V + 3

. The equation is

\frac{0.55V + 3}{V + 3} = 0.58

. Multiplying both sides by

V + 3

yields

0.55V + 3 = 0.58(V + 3) \implies 0.55V + 3 = 0.58V + 1.74

. Rearranging terms gives

3 - 1.74 = 0.58V - 0.55V \implies 1.26 = 0.03V \implies V = 42

Key Concept

Solving multi-step mixture word problems using equations involving fractions, decimals, and percentages.

Alternative Method

Instead of variables for the total volume, solve the problem by tracking parts. Initially, the mixture consists of

20\%

Solution X,

25\%

Solution Y, and

55\%

Solution Z. The ratio of the volume of Solution Z to the combined volume of Solution X and Solution Y is

55 : 45

, which simplifies to

11 : 9

. Let the volume of X and Y combined be

9x

liters, and the initial volume of Z be

11x

liters, making the initial total volume

20x

liters. Adding

3.0

liters of Solution Z does not change the combined volume of X and Y (

9x

liters). In the new mixture, Solution Z constitutes

58\%

, meaning the combined volume of X and Y must constitute

100\% - 58\% = 42\%

of the new mixture. Thus, the new total volume is

9x / 0.42 = 150x / 7

liters. The difference between the new and original total volumes is the

3.0

liters added:

(150x / 7) - 20x = 3 \implies 10x / 7 = 3 \implies x = 2.1

. The original total volume was

20x = 20(2.1) = 42

liters.

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic