Problem 53 - Algebra and Trigonometry

Question

A chemist has two large containers of sulfuric acid solution, with different concentrations of acid in each container. Blending 300 mL of the first solution and 600 mL of the second gives a mixture that is 15\(\%\) acid, whereas 100 mL of the first mixed with 500 mL of the second gives a 12\(\frac{1}{2} \%\) acid mixture. What are the concentrations of sulfuric acid in the original containers?

Step-by-Step Solution

Answer

The concentrations are 25\(\%\) in the first container and 10\(\%\) in the second container.

Step 1: Represent the Unknowns

Let the concentration of sulfuric acid in the first container be \( x \% \), and in the second container be \( y \% \). These are the unknowns that we need to find.

Step 2: Formulate the First Equation

For the blend of 300 mL from the first and 600 mL from the second container, which results in a 15\(\%\) acid solution: \[ 0.15 \times (300 + 600) = 300 \times \frac{x}{100} + 600 \times \frac{y}{100} \] Simplify the equation to get: \[ 135 = 3x + 6y \] This equation simplifies further to: \[ 3x + 6y = 135 \] or \[ x + 2y = 45 \]

Step 3: Formulate the Second Equation

For the mix of 100 mL from the first and 500 mL from the second container, resulting in a 12\(\frac{1}{2}\%\) acid solution: \[ 0.125 \times (100 + 500) = 100 \times \frac{x}{100} + 500 \times \frac{y}{100} \] Simplify this equation to get: \[ 75 = x + 5y \]

Step 4: Solve the System of Equations

We have two equations now: \[ \begin{align*} x + 2y &= 45 \ x + 5y &= 75 \end{align*} \] Subtract the first equation from the second: \[ (x + 5y) - (x + 2y) = 75 - 45 \] \[ 3y = 30 \] \[ y = 10 \] Substitute \( y = 10 \) back into the first equation: \[ x + 2(10) = 45 \] \[ x + 20 = 45 \] \[ x = 25 \]

Step 5: Verify the Solution

Substitute \( x = 25 \) and \( y = 10 \) back into both equations to check correctness: - For the first equation: \[ x + 2y = 25 + 2(10) = 45 \] Correct.- For the second equation: \[ x + 5y = 25 + 5(10) = 75 \] Correct.

Key Concepts

System of Equations

In algebra, a system of equations is a set of two or more equations that have common variables. Solving a system of equations involves finding the values of the variables that satisfy all equations simultaneously.
In the context of our sulfuric acid problem, we have two unknowns: the concentration of sulfuric acid in each of the two containers. Each blending scenario provides us with an equation.
Here are the key steps to solve systems:

• Identify the unknowns (variables) in your problem.
• Develop equations based on the scenario descriptions.
• Use various methods like substitution or elimination to solve these equations.

For our exercise, after setting up the equations from the problem, we used the elimination method. This approach involved manipulating the equations to cancel out one of the unknowns, making it easier to solve for the other.

Concentration Problems

Concentration problems are a classic in algebra, especially in chemistry or similar fields. They involve determining the concentration (often in percentage) of substances in various mixtures.
In this exercise, we encountered a common concentration scenario: mixing two solutions to form mixtures with specific acid concentrations. The core task was to find out how concentrated the original solutions were.
When faced with these problems:

• Convert percentage concentrations to decimals to simplify calculations. For example, a 15% solution becomes 0.15.
• Consider the total volume of the solution when forming your equations.
• Account for the contribution of each solution to the overall mixture's concentration.

Mastering concentration problems requires practice and a strong understanding of the relationship between volume and concentration.

Linear Equations

Linear equations are foundational in algebra, describing relationships in which the variables are of the first degree (power of one). They typically appear in the form: \[ ax + by = c \]
In our exercise, each relationship between the volumes and concentrations is represented by a linear equation. Solving such equations is central to resolving the problem's query.
Key points to remember with linear equations:

• Their graphs are straight lines.
• They often emerge from proportional relationships and direct variability as seen when solutions are mixed.
• Being familiar with techniques such as balancing, cross-multiplying, and using graphing methods can be very useful.

Thanks to the linear nature of our equations, once they're formulated, they shed light on the relationship between the variables, helping us swiftly reach our solution.

Mixture Problems

Mixture problems in math involve combining substances to achieve a desired outcome, often regarding concentration or quantity. They are common in real-world scenarios like chemistry or cooking.
In the given exercise, we combined different volumes of sulfuric acid solutions to achieve a particular concentration, creating a classical mixture problem. Here's how to approach them:

• Use known quantities (either volume or mass) to develop equations representing the scenario.
• Account for how much each component contributes to the mixture.
• Reconcile all equations to find the unknown variable values.

Mixture problems test your ability to juggle multiple variables and their contributions to the whole, offering practical applications for mathematical concepts in everyday life.