Problem 58 - Algebra for College Students

Question

How many gallons of a \(12 \%-\) salt solution must be mixed with 6 gallons of a \(20 \%\)-salt solution to obtain a \(15 \%\)-salt solution?

Step-by-Step Solution

Answer

Mix 10 gallons of the 12% salt solution.

Step 1: Define Variables

Let's define the variable for our use. Let \( x \) be the number of gallons of the \(12\%\) salt solution we need to find.

Step 2: Set Up the Equation

The amount of salt in an \(x\) gallon \(12\%\) solution is \(0.12x\) and the amount of salt in the \(6\) gallon \(20\%\) solution is \(0.20 \times 6 = 1.2\). The total amount of salt after mixing should give a \(15\%\) concentration. Hence, the equation is:\[ 0.12x + 1.2 = 0.15(x + 6) \]

Step 3: Simplify the Equation

Expand the right-hand side of the equation: \[ 0.15(x + 6) = 0.15x + 0.9 \]So the equation becomes:\[ 0.12x + 1.2 = 0.15x + 0.9 \]

Step 4: Solve for x

Rearrange the equation to group \(x\) terms on one side:\[ 1.2 - 0.9 = 0.15x - 0.12x \]This simplifies to:\[ 0.3 = 0.03x \]Solve for \(x\):\[ x = \frac{0.3}{0.03} = 10 \]

Step 5: Interpret the Solution

We found that \(x = 10\). This means you need to mix \(10\) gallons of the \(12\%\) salt solution with \(6\) gallons of the \(20\%\) salt solution to get a \(15\%\) salt solution.

Key Concepts

Percentage Concentration

In mixture problems like these, percentage concentration is a key concept. It indicates how much of a certain substance (like salt) is present within a solution. This concentration is shown as a percentage of the whole solution. For instance, a 12% salt solution means that 12% of the solution's total volume is salt. This idea is vital when mixing different solutions together, since it helps predict the final concentration of the mix.

When dealing with percentages, remember these points:

• The percentage concentration is always relative to the entire volume of the solution.
• It helps to convert the percentage into decimals for calculations; for example, 12% becomes 0.12.
• Understanding how to find the amount of solute (like salt) using given percentages is crucial. For example, in a solution where the concentration is 0.12, multiplying this by the total volume gives the amount of salt.

When you're tasked with mixing solutions to achieve a specific concentration, knowing these basics is essential.

Mixture Problems

Mixture problems are a common type of algebra word problem. They involve combining different substances to reach a desired outcome, like a certain concentration. These problems help you understand how different quantities affect each other, and they require careful setup of equations to solve.

Key aspects of mixture problems include:

• Identifying the known quantities, such as the initial concentrations and volumes of the solutions.
• Determining the desired concentration and total volume of the final mixture.
• Setting up equations based on these knowns, which reflect the total amount of substance (e.g., salt) in the final solution.

By systematically approaching the problem, you can find out how much of each component you need to achieve your goal. In the provided exercise, the problem was to find how many gallons of a 12%-salt solution would be needed to make a 15% solution when mixed with a 20%-salt solution. It's a good example of how key characteristics of the components guide the required mixture.

Equation Solving

Solving equations is a fundamental part of tackling algebra word problems, especially in mixture examples like this one. Once you have set up the equation based on the given information, the next step is to solve it correctly.

Here’s how you can solve such equations:

• First, write down the equation representing the problem. For our mixture, this was \(0.12x + 1.2 = 0.15(x + 6)\).
• Next, simplify both sides as needed, such as expanding \(0.15(x + 6)\) to get \(0.15x + 0.9\).
• Rearrange the equation to get all terms with the unknown variable on one side and constants on the other, in this case leading to \(0.3 = 0.03x\).
• Finally, solve for the variable by dividing through by the coefficient of \(x\), resulting in \(x = 10\).

This logical progression is typical in algebra. It's crucial to check your work to ensure your solution makes sense in the context of the problem. The understanding of each step is vital to not only finding the right answer but also to gaining deeper insights into algebraic methods.