A chemist is preparing 120\text{ milliliters} of a chemical mixture with an overall acid c… — ACT

A chemist is preparing $120\text{ milliliters}$ of a chemical mixture with an overall acid concentration of $27.5\%$ by volume. To do this, she mixes Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $25\%$ acid by volume), and Solution $Z$ ( $40\%$ acid by volume). The chemist decides that the volume of Solution $Y$ must be exactly $20\text{ milliliters}$ less than twice the volume of Solution $X$ used in the mixture. What is the positive difference, in milliliters, between the volume of Solution $Z$ and the volume of Solution $X$ in the final mixture?

Cevap: 20 milliliters

Cevap

The positive difference between the volume of Solution Z and Solution X is 20 milliliters.

The correct answer is 20, because solving the system of equations yields a volume of 30 milliliters for Solution X and 50 milliliters for Solution Z. The positive difference between these two volumes is 20 milliliters.

Adım Adım Çözüm

Define variables for the volume of each solution used in the mixture.

Let

x

be the volume of Solution

X

y

be the volume of Solution

Y

, and

z

be the volume of Solution

Z

(all in milliliters).

Defining variables allows for translating the word problem's conditions into algebraic equations.

Translate the given information into a system of three linear equations.

Equation 1 (Total Volume):

x + y + z = 120

. Equation 2 (Total Acid):

0.10x + 0.25y + 0.40z = 33

(since

27.5\%

120\text{ ml}

33\text{ ml}

). Equation 3 (Volume Relation):

y = 2x - 20

These equations model the constraints and quantities described in the problem.

Express

z

in terms of

x

by substituting the expression for

y

into the total volume equation.

x + (2x - 20) + z = 120 \implies 3x - 20 + z = 120 \implies z = 140 - 3x

This substitution reduces the system to two variables (

x

and

z

), making it easier to solve.

Substitute the expressions for

y

and

z

in terms of

x

into the acid equation and solve for

x

0.10x + 0.25(2x - 20) + 0.40(140 - 3x) = 33 \implies 0.10x + 0.50x - 5 + 56 - 1.20x = 33 \implies -0.60x + 51 = 33 \implies -0.60x = -18 \implies x = 30

This isolates the single variable

x

so that its value can be calculated.

Determine the volume of Solution

Z

and calculate the final positive difference.

z = 140 - 3(30) = 50

. The positive difference between the volume of Solution

Z

and Solution

X

|50 - 30| = 20

This directly answers the question's requirement for the difference between the two solution volumes.

Anahtar Kavram

Translating word problems into a system of three linear equations and solving them using substitution.

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Adım Adım Çözüm

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