Q 4.54 - Intermediate Algebra

Question

Translate to a system of equations and solve:

Anatole needs to make $250$ milliliters of a $25\%$ solution of hydrochloric acid for a lab experiment. The lab only has

a $10\%$ solution and a $40\%$ solution in the storeroom. How much of the $10\%$ and how much of the $40\%$ solutions

should he mix to make the $25\%$ solution?

Step-by-Step Solution

Answer

The volume of $10\%$ that should be mixed is $125 ml$

The volume of $40\%$ solution that should be mixed is $125 ml$

Step 1. Given information

• The volume of solution needed is $250 ml$ of $25\%$ of hydrochloric acid.
• The concentration we have is $10\%$ and $40\%$ of hydrochloric acid.

Step 2. Form the equations

Let

$$\begin{gathered} x = number of ml of 10\% solution. \\ y = number of ml of 40\% solution \end{gathered}$$

So, according to the question, 

$x+y=250\_\_\_\left(1\right)$

The total volume of hydrochloric acid at $25\%$ is:

$$\begin{gathered} =250\times 25\% \\ =250\times 0.25 \\ =62.5 \end{gathered}$$

So again according to the question.

$0.10x+0.40y=62.5\_\_\_\left(2\right)$

Step 3. Solve the equations by elimination method

Multiplying the equation $\left(1\right)$ by $0.10$, we get:

$$\begin{gathered} 0.10(x+y)=0.10\times 250 \\ 0.10x+0.10y=25\_\_\_\left(3\right) \end{gathered}$$

Subtracting equation $\left(3\right)$ from $\left(2\right)$ , we get:

$$\begin{gathered} 0.10x+0.40y-(0.10x+0.10y)=62.5-25 \\ 0.30y=37.5 \end{gathered}$$

Dividing both sides by $0.30$, we get:

$$\begin{gathered} \frac{0.30y}{0.30}=\frac{37.5}{0.30} \\ y=125 \end{gathered}$$

Step 4. Find the value of x

Substituting the value of $y$ in the equation $\left(1\right)$, we get:

$$\begin{gathered} x+y=250 \\ x+125=250 \\ x+125-125=250-125 \\ x=125 \end{gathered}$$

So, Anatole should mix $125 ml$ of $10\%$ solution and $125 ml$ of  $40\%$ solution.