Digital SAT Question: Mixing Problems

Mixing problems are a type of math problem that involves finding the concentration of a substance in a mixture. These problems can be challenging, but they can be solved using a step-by-step approach.

Key Concepts

• Concentration: The concentration of a substance is the amount of that substance that is present in a given volume of solution. Concentration can be expressed in different units, such as molarity (M), parts per million (ppm), or percentage (%).
• Mixture: A mixture is a combination of two or more substances that are not chemically bonded to each other. The substances in a mixture can be in different phases, such as solid, liquid, or gas.

Step-by-Step Approach

To solve mixing problems, follow these steps:

1. Identify the given information: Determine the concentration and volume of each of the substances in the mixture.
2. Calculate the total volume of the mixture: Add the volumes of all the substances in the mixture.
3. Calculate the total amount of substance: Multiply the concentration of each substance by its volume.
4. Calculate the concentration of the mixture: Divide the total amount of substance by the total volume of the mixture.

Examples

Example 1:

A chemist mixes 50 mL of a 1 M solution of sodium chloride with 100 mL of a 0.5 M solution of sodium chloride. What is the concentration of the final mixture?

Solution:

1. Identify the given information:

   • Concentration of solution 1: 1 M
   • Volume of solution 1: 50 mL
   • Concentration of solution 2: 0.5 M
   • Volume of solution 2: 100 mL

2. Calculate the total volume of the mixture:

   

   • Total volume = 50 mL + 100 mL = 150 mL

3. Calculate the total amount of sodium chloride:

   • Total amount = (1 M x 50 mL) + (0.5 M x 100 mL) = 100 mmol

4. Calculate the concentration of the mixture:

   

   • Concentration = 100 mmol / 150 mL = 0.67 M

Example 2:

A hospital needs to prepare a 5% glucose solution. How many grams of glucose are needed to prepare 250 mL of the solution?

Solution:

1. Identify the given information:

   • Concentration of solution: 5% (w/v)
   • Volume of solution: 250 mL

2. Calculate the mass of glucose needed:

   • First, convert concentration to g/mL: 5% = 5 g / 100 mL
   • Mass of glucose = Concentration x Volume = 5 g / 100 mL x 250 mL = 12.5 g

Tips and Tricks

• Check your units carefully: Make sure the units of concentration and volume are consistent throughout the problem.
• Use dimensional analysis: Set up the equation so that the units cancel out and give you the desired result.
• Don’t be afraid to make assumptions: If the problem does not provide all the necessary information, make reasonable assumptions based on common knowledge or the context of the problem.

Conclusion:

Mixing problems are a common type of SAT math problem. By understanding the key concepts and following a step-by-step approach, students can solve these problems accurately and efficiently.

Additional Resources

• Khan Academy: Mixing Problems
• Varsity Tutors: Mixing Problems
• The Princeton Review: Mixing Problems