Problem 57 - Algebra for College Students

Question

Solve each problem. How many liters of a \(10 \%\) acid solution must be mixed with \(10 \mathrm{~L}\) of a \(4 \%\) solution to obtain a \(6 \%\) solution?

Step-by-Step Solution

Answer

5 liters of the \(10 \%\) acid solution are needed.

Step 1 - Define Variables

Let's define the unknown quantity as follows: let \( x \) be the number of liters of the \(10 \%\) acid solution that we need to mix.

Step 2 - Write Down the Concentration Equation

We need to find the amount of acid in each solution using the percentage concentration. The amount of acid in the \( x \) liters of \(10 \%\) solution is \(0.1x\) liters. The amount of acid in \(10 \, \text{L}\) of \(4 \%\) solution is \(0.04 \times 10 = 0.4\) liters. The final solution will have a volume of \((x + 10) \) liters with a \( 6 \%\) concentration.

Step 3 - Set Up the Equation

To create the \(6 \% \) solution, the total amount of acid in the final solution should be equal to \(6 \% \) of the volume of the final solution. The equation is: \[0.1x + 0.4 = 0.06(x + 10)\].

Step 4 - Solve the Equation

First, distribute \(0.06\) on the right side: \[0.1x + 0.4 = 0.06x + 0.6\]. Then combine like terms: \[0.1x - 0.06x = 0.6 - 0.4\]. This simplifies to: \[0.04x = 0.2\]. Finally, solve for \(x\) by dividing both sides by 0.04: \[x = \dfrac{0.2}{0.04} = 5\].

Key Concepts

Acid Concentration

Acid concentration refers to the proportion of acid present in a solution. This is usually expressed as a percentage. For instance, a 10% acid solution means that 10% of the solution's volume is pure acid, and the remaining 90% is another substance (typically water). Understanding concentrations is vital for mixture problems because it allows us to calculate the amount of pure substance when different solutions are mixed.
In the given example, the 10% acid solution has more acid per liter than the 4% solution. When you mix these, you aim to achieve a solution with the desired concentration, in this case, 6%.

Solving Equations

In mixture problems, solving equations is used to find the unknown quantity. We set up an equation that reflects the mixing process.
1. Define a variable for the unknown quantity, such as the liters of acid solution needed.
2. Write down the concentration equation, which includes the amounts of each component in the solutions.
3. Set up the equation based on the total volume and concentration desired.
For this exercise, we defined 'x' as the liters of 10% solution needed. We then formulated the equation based on the concentration of acid from all solutions. The general equation in this context will balance the amount of pure acid in the solution to match the required concentrations. Eventually, simple algebraic manipulation helps find the value of 'x'.

Percentage Solutions

Percentage solutions state the concentration of a substance in a mixture. This is important for understanding how much of a component (like acid) is present in a solution.
When solving mixture problems, it's crucial to convert these percentages into actual quantities. We can then use these quantities to set up equations.
For instance, a 4% solution means there are 4 parts of acid in every 100 parts of the solution. So, in 10 liters, there would be 0.4 liters of acid.
Similarly, a 10% solution has 10 parts of acid in every 100 parts. Converting these verbal descriptions into mathematical terms allows us to combine and manipulate the quantities to find the desired concentration in the final mixture.