Problem 64 - Intermediate Algebra

Question

If a mixture is to be made from solutions with concentrations of \(12 \%\) and \(30 \%,\) can the mixture have a concentration less than \(12 \% ?\) Can the mixture have a concentration greater than \(30 \% ?\) Explain.

Step-by-Step Solution

Answer

The mixture cannot have a concentration less than 12% or greater than 30%.

Step 1: Understanding the Problem

We have two solutions with concentrations of 12% and 30%. We need to determine if a mixture can have a concentration less than 12% or greater than 30%.

Step 2: Concept of Mixing Solutions

When mixing solutions of different concentrations, the resultant concentration will always lie between the two original concentrations unless external substances are added.

Step 3: Analyzing Lower Limit

The lowest possible concentration from the mixture of these two solutions without adding another solution is 12%. To achieve less than this, you would need a solution with a concentration less than 12%, which is not present.

Step 4: Analyzing Upper Limit

The highest possible concentration from mixing these two solutions without adding another solution is 30%. To achieve greater than this, you would need a solution with a concentration greater than 30%, which is not present.

Step 5: Conclusion

Since only solutions with concentrations of 12% and 30% are available, the mixture cannot have a concentration less than 12% or greater than 30%.

Key Concepts

Solution Mixing

When we talk about solution mixing, we're combining liquid solutions with different concentrations to find a middle ground. Think of it as merging the strengths of two options into one. For example:

• The first solution might be 12% sugar, and the second one might be 30% sugar.
• We're aiming to create one solution that balances somewhere between the two.

The process of mixing is akin to creating a music playlist by combining songs of different tempos to strike the perfect harmony. The key here is that unless we add something entirely different, like pure water or a new solution, our result will always fall within the range of the initial concentrations. It's like making sure your music is not too fast or too slow, but just right!

Mixture Concentration Limits

Every time we mix solutions, there are limits defined by the properties of those very solutions. These mixture concentration limits are the boundaries that the concentration of the final mixture cannot exceed.

• If you were to mix a 12% and a 30% solution, the mixture's concentration must stay between 12% and 30%.
• If you want a concentration lower than 12%, you'd need to mix in a solution with a lesser percentage, like water, which is 0% concentration. Yet, in this case, that's not part of our current setup.

Moreover, it would be impossible to get a concentration above 30% without incorporating a stronger solution. Imagine trying to make lemonade taste more lemony without adding more lemons! Thus, the final product can't be weaker or stronger than the existing solutions.

Chemical Solutions

A chemical solution is a homogeneous mixture composed of two or more substances. When we delve into chemical solutions, we are stepping into the realm of chemistry where solutions behave according to certain expectations.

• The solvent is the player that dissolves, usually present in the greater amount like water in sugar water.
• The solute is what gets dissolved, such as sugar.

The beauty of chemical solutions lies in their predictability — no unexpected surprises if we stick to the rules! In our example, we had solutions of different sugar concentrations. When mixed together without interference, they form a new solution whose concentration is predictably between the two. It's much like blending paints; mix blue and yellow, and you predictably get green, not something out of the blue, pun intended! The key takeaway is the understanding of limits and behavior in solution mixing — these help predict the outcome accurately.