Problem 47 - Algebra 1

Question

in your chemistry class you have a bottle of 5\% boric acid solution and a bottle of \(2 \%\) boric acid solution. You need 60 milliliters a \(3 \%\) boric acid solution for an experiment. How much of each solution do you need to mix together?

Step-by-Step Solution

Answer

The volumes of 5\% and \(2 \%\) boric acid solutions required for preparing the \(3 \%\) boric acid solution will be determined by solving the system of equations

Step 1: Set up the volume equation

Let x be the volume of 5\% boric acid solution and y be the volume of \(2 \%\) boric acid solution needed. Since the total volume needed is 60 mL, the equation will be: \(x + y = 60\)

Step 2: Set up the boric acid concentration equation

Since 5\% of the solution x and \(2 \%\) of the solution y should compose \(3 \%\) of the total solution, the equation will be: \(0.05x + 0.02y = 0.03 * 60\)

Step 3: Solve the system of equations

This system of equations can be solved by substitution or elimination method. In this case, solve the first equation for x: \(x = 60 - y\), and substitute into the second equation to find the value of y: \(0.05(60-y) + 0.02y = 1.8\)

Step 4: Calculate the volume of each solution

Calculate the value of y and then substitute it back into \(x = 60 - y\) equation to find the value of x. This will provide the volume of each solution needed to prepare a \(3 \%\) boric acid solution

Key Concepts

System of Equations

When faced with a problem where you have different variables to solve, a system of equations is your best friend. In this case, we need to determine how much of two different solutions should be mixed to achieve a desired concentration. This involves setting up equations based on the given conditions and solving them together.
A system of equations often consists of two or more equations, which share common variables. For example, if you have:

• Equation 1: This represents the total volume requirement, like in our problem where the sum of the volumes of the two solutions equals 60 mL: \(x + y = 60\).
• Equation 2: This represents the concentration equation, linking the percentages to the desired overall concentration: \(0.05x + 0.02y = 1.8\).

The goal is to find the values of these variables that satisfy both equations at once. Methods commonly used include substitution, where you solve one equation for one variable, then substitute into the other, or elimination, where you try to cancel out one variable by adding or subtracting the equations.

Concentration Problems

Concentration problems in algebra involve finding the right mix of components to achieve a particular strength of a solution, as seen in this exercise where different boric acid solutions must be combined to reach a 3% concentration.
Understanding concentration is crucial: it's the amount of solute (in this case, boric acid) divided by the total volume of the solution, often expressed as a percentage. This tells you how strong the solution is.
To solve concentration issues, set up an equation that represents this idea. For example, if you need 3% of a 60 mL solution to be boric acid, you are essentially saying you need: \[ 0.03 \times 60 = 1.8 \] milliliters of boric acid in total. Your task is then to determine how much of each available solution will contribute to this total, leading to the effective concentration equation: \(0.05x + 0.02y = 1.8\). This equation ensures that the resulting mixture will meet the required concentration.

Mixture Problems

Mixture problems in algebra are puzzles that ask you to blend different components to produce a final product with specified properties. Here, you're dealing with two different boric acid solutions of varying concentrations—5% and 2%—and the task is to create a 3% solution from these.
The essence of solving mixture problems is to apply the right proportions. It's like making a recipe in the kitchen, where each ingredient must perfectly balance with the others to ensure the desired taste and texture.
In this exercise, the setup involves two main ideas: one is the total volume equation \(x + y = 60\), and the other is the concentration equation \(0.05x + 0.02y = 1.8\). Once you lay out these equations, the problem is about mixing the right volumes of each of the available solutions so that combined, they deliver exactly the solution you need (60 mL of 3% boric acid). In these mixture problems, the relationships and interactions are just as crucial as the individual quantities of each component.