Problem 106 - College Algebra Essentials

Question

Suppose that \(x\) liters of pure acid are added to 200 liters of a \(35 \%\) acid solution. a. Write a formula that gives the concentration, \(C,\) of the new mixture. (Hint: See Exercise \(105 .\) ) b. How many liters of pure acid should be added to produce a new mixture that is \(74 \%\) acid?

Step-by-Step Solution

Answer

To produce a new mixture that is 74% acid, 300 litres of pure acid should be added to the initial solution.

Step 1: Derive the formula for the concentration of acid in the mixture

To derive the concentration of the acid in the mixture, you need to account for the amount of acid present before and the acid being added. Initially, you have a 200 litres of 35% acid solution. This means you have \(200 \times 0.35 = 70\) litres of acid in the initial solution. When you add \(x\) litres of pure acid to it, the total volume of the mixture becomes \(200 + x\) litres and the total amount of acid becomes \(70 + x\) litres. The concentration, \(C\), can thus be represented as the total amount of acid (70+ x) divided by the total volume of the resulting solution (200+ x), that is, \(C = \frac{(70 + x)}{(200 + x)}\)

Step 2: Calculate how many litres of pure acid should be added to produce a 74% acid mixture

Once you have the formula for \(C\), you can equate it to 0.74 (the desired concentration) and solve for \(x\) to determine how many litres of pure acid you should add. Setting \(\frac{(70 + x)}{(200 + x)} = 0.74\) and solving for \(x\) gives \(x = \frac{(0.74 \times 200 - 70)}{(1 - 0.74)}\)

Step 3: Compute the result

Substitute the known values into the equation to find the value of \(x\), which results in \(x = \frac{(148 - 70)}{(0.26)} = 300\) litres. Thus you need to add 300 litres of pure acid to the initially present 200 litres acid solution to end up with a 74% acid mixture.

Key Concepts

Understanding Algebraic Expressions in Mixture Problems

When solving mixture problems, algebraic expressions play a crucial role. They help us represent mixing scenarios mathematically. In our exercise, an algebraic expression was formed to calculate the concentration of acid in the mixture. This begins by defining variables that reflect the quantities involved, such as the volume of the initial solution and the amount of pure acid added.
Here is how it works:

• Define the variables: Let the initial amount of the acid solution be denoted by the known quantity, here 200 liters, and the unknown, say \(x\), for the liters of pure acid added.
• Establish relationships between variables: The algebraic expression reflects the logical relationship between these variable quantities. For example, the total amount of acid in the solution after mixing is expressed as \(70 + x\), where 70 liters is the initial amount of acid.
• Formulate the concentration: This is represented through an algebraic fraction \(C = \frac{(70 + x)}{(200 + x)}\).

By setting up such expressions, you transform verbal descriptions into mathematical statements, making it easier to perform computations and derive further insights.

Calculating Concentrations in Mixture Problems

Concentration is a measure of how much of one substance is present in a mixture. In terms of solutions, it's usually expressed as a percentage of the total volume. Concentration calculation is about understanding and computing how much of one component, say acid, is in comparison to the whole mixture.
For example, let's calculate the concentration of the new acid mixture:

• Original amount of acid: In our case, this was 70 liters (from 200 liters of a 35% acid solution).
• Add additional pure acid: We then add \(x\) liters of pure acid, where pure means 100% concentration.
• Find the concentration: Now, we use the formula \(C = \frac{(70 + x)}{(200 + x)}\) to find the concentration \(C\) of the new resulting solution.

By following these steps, concentration calculations help us understand the composition of our resulting mixtures and make predictions about mixture properties, such as the effectiveness of solutions in chemical reactions.

Solving Percentage Problems

Percentage problems often appear in mixture calculations as we are usually interested in the concentration expressed in percentage. To solve percentage problems like the one in this exercise, understanding how percentages relate to fractions is vital.
Here is how percentage problems are tackled:

• Translate percentages into decimals: For example, a 35% solution translates to 0.35 when expressed as a decimal.
• Apply algebra to solve for unknowns: With the formula \(\frac{(70 + x)}{(200 + x)} = 0.74\), we set it equal to the desired concentration percentage in decimal form, which is 0.74 for a 74% solution.
• Solve using basic equation-solving techniques: We isolate \(x\) to determine how much more of the component is needed to reach our target concentration.

Solving these percentage problems involves straightforward arithmetic once you establish the relationship between current and desired states of your mixtures.