Problem 12 - Elementary Algebra Within Reach

Question

Chemistry You need 175 milliliters of a \(6 \%\) hydrochloric acid solution for an experiment. Your chemistry lab has a bottle of \(3 \%\) hydrochloric acid solution and a bottle of \(10 \%\) hydrochloric acid solution. How many milliliters of each solution should you mix together?

Step-by-Step Solution

Answer

You should mix 75 milliliters of the \(3\%\) hydrochloric acid solution with 100 milliliters of the \(10\%\) hydrochloric acid solution.

Step 1: Define variables

Let's say the volume in milliliters of the \(3\%\) solution used is represented by \(x\) and the \(10\%\) solution used is represented by \(y\).

Step 2: Set up the equations

The total volume should equal 175 milliliters, hence we have \(x + y = 175\). Furthermore, the total hydrochloric acid concentration should sum up to \(6\%\). Thus the second equation is: \(0.03x + 0.1y = 175*0.06 = 10.5\)

Step 3: Solve the system of equations

This system can be solved by substitution or elimination method, here we use substitution. From the first equation we can get \(y = 175 - x\) . Substituting \(y\) in second equation we get \(0.03x + 0.1(175-x) = 10.5\). Solving this we get \(x = 75\). Substituting \(x = 75\) in \(y = 175 - x\) we get \(y = 100\) .

Step 4: Interpret the solution

The solution \(x = 75, y = 100\) indicates that 75 milliliters of the \(3\%\) hydrochloric acid solution and 100 milliliters of the \(10\%\) hydrochloric acid solution are to be used to achieve desired mixture.

Key Concepts

System of Equations

In many algebraic word problems, especially in mixture and concentration problems, you need to establish a system of equations. A system of equations consists of two or more equations that share the same set of unknowns. In this particular problem, we set up two equations since we have two different variables to consider: the volumes of two different solutions that combine to create a desired concentration.

• The first equation comes from the total volume requirement: \(x + y = 175\), where \(x\) and \(y\) are the volumes of the solutions.
• The second equation ensures the correct concentration: \(0.03x + 0.1y = 10.5\). This equation represents the sum of the concentrations from both solutions, which should equal the desired concentration of the final mix.

Each equation represents a linear relationship between the variables, and solving these equations simultaneously gives us a solution that satisfies both conditions. This method is essential for solving many real-world problems where two conditions need to be met together.

Substitution Method

To solve the system of equations from the word problem, we can use various methods, and one popular approach is the substitution method. This involves expressing one variable in terms of the other, allowing us to focus on a single equation to find a numerical value.

Here’s how this works in the example problem:

• From the equation \(x + y = 175\), solve for one variable: \(y = 175 - x\).
• Substitute this expression for \(y\) in the second equation: \(0.03x + 0.1(175-x) = 10.5\).
• This allows us to solve for \(x\) directly.
• Once we have \(x\), substitute it back into \(y = 175 - x\) to find \(y\).

Substitution is particularly useful when one of the equations is easily solvable for one variable. It keeps the algebra straightforward and reduces error potential, especially useful for beginners learning how to approach algebraic equations.

Concentration Mixture Problems

Concentration mixture problems require us to combine solutions of different concentrations to achieve a blend with the desired concentration. This type of problem is common in chemistry, where different strengths of solutions need mixing.

To solve these problems, follow these basics steps:- Understand what concentrations are involved (in this case, \(3\%\) and \(10\%\)) and what the desired concentration is (\(6\%\)).- Establish equations based on both the total volume needed and the total concentration required. This ensures both the physical quantity and the strength of the solution are correct.- Use a system of equations, as outlined above, to determine the volumes of each solution to mix.

Concentration problems test your ability to interpret and create mathematical models from word problems. The algorithms to solve them rely heavily on basic algebraic skills but also introduce practical reasoning skills applied in various scientific contexts.