A chemist needs to mix a 10\% acid solution with a 30\% acid solution to create a 100-mill… — SAT

A chemist needs to mix a $10\%$ acid solution with a $30\%$ acid solution to create a $100$ -milliliter mixture. If the final mixture must be $18\%$ acid, how many milliliters of the $10\%$ acid solution should the chemist use?

A
40
60Answer
C
70
D
80

Answer

60 milliliters

The correct answer is 60 milliliters. Defining

x

as the volume of the

10\%

solution and

y

as the volume of the

30\%

solution gives the system of equations

x + y = 100

and

0.10x + 0.30y = 18

. Substituting

y = 100 - x

into the second equation yields

0.10x + 30 - 0.30x = 18

. Simplifying this equation gives

-0.20x = -12

, which simplifies to

x = 60

Step-by-Step Solution

Define variables for the volume of each solution and set up the system of equations representing the total volume and the total amount of pure acid.

Let

x

be the number of milliliters of the

10\%

acid solution, and let

y

be the number of milliliters of the

30\%

acid solution. The system is:

\begin{aligned} x + y &= 100 \\ 0.10x + 0.30y &= 18 \end{aligned}

To represent the physical relationships between the two solutions mathematically.

Solve the first equation for

y

in terms of

x

and substitute this expression into the second equation.

y = 100 - x

0.10x + 0.30(100 - x) = 18

To reduce the system to a single linear equation in terms of

x

Distribute the coefficients, combine like terms, and solve for

x

0.10x + 30 - 0.30x = 18

-0.20x + 30 = 18

-0.20x = -12

x = 60

To isolate

x

and determine the volume of the

10\%

acid solution.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination.

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