A chemist wants to create 100 milliliters of a 42.5\% acid solution by mixing three differ… — ACT

A chemist wants to create $100$ milliliters of a $42.5\%$ acid solution by mixing three different acid solutions: a $10\%$ acid solution, a $20\%$ acid solution, and an $80\%$ acid solution. She decides that the volume of the $20\%$ acid solution used must be exactly $3$ times the volume of the $10\%$ acid solution used. What is the difference, in milliliters, between the volume of the $80\%$ acid solution and the volume of the $10\%$ acid solution used in the final mixture?

Answer: 25 milliliters

Answer

The difference between the volume of the

80\%

acid solution and the volume of the

10\%

acid solution is

25

milliliters.

Solving the system of equations yields that

15

milliliters of the

10\%

solution,

45

milliliters of the

20\%

solution, and

40

milliliters of the

80\%

solution are needed. The difference between the volume of the

80\%

solution and the

10\%

solution is

40 - 15 = 25

milliliters.

Step-by-Step Solution

Define variables for the volume of each acid solution.

Let

x

be the volume of the

10\%

solution,

y

be the volume of the

20\%

solution, and

z

be the volume of the

80\%

solution.

This establishes algebraic representations for the unknowns.

Set up a system of linear equations based on the relationships given in the problem statement.

x + y + z = 100

(total volume)

y = 3x

(relationship between the

20\%

and

10\%

solutions)

0.10x + 0.20y + 0.80z = 42.5

(total acid content)

Translating word problems to mathematical equations allows us to solve for the variables systematically.

Reduce the system to a single equation in terms of

x

by substituting

y = 3x

and expressing

z

in terms of

x

4x + z = 100 \implies z = 100 - 4x

Substitute both into the acid equation:

0.10x + 0.20(3x) + 0.80(100 - 4x) = 42.5

Substitution simplifies the system of equations to a single linear equation with one variable.

Solve the simplified linear equation for

x

0.70x + 80 - 3.20x = 42.5 \implies -2.50x = -37.5 \implies x = 15

This determines the volume of the

10\%

acid solution.

Calculate the volume of the

80\%

solution,

z

z = 100 - 4(15) = 40

This determines the volume of the

80\%

acid solution.

Find the difference between

z

and

x

z - x = 40 - 15 = 25

The question asks for the difference between these two volumes.

Key Concept

Translating and solving systems of linear equations from verbal descriptions (mixture problems).

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic