solve mixture priblems whith proportions quick check

Answer 1

To solve mixture problems involving proportions, you can use a systematic approach. Here are some steps and a quick example to illustrate how to solve such problems.

Steps to Solve Mixture Problems

1. Define Variables: Assign variables to the quantities or parts of the mixture you need to find.

2. Set Up Proportions: Use the information given about the proportions of the components in the mixture and how they will combine.

3. Write an Equation: If applicable, write an equation based on the total quantities, or the specific proportions required.

4. Solve the Equation: Use algebra to solve the equation for the unknown variable.

5. Check Your Work: Verify your answer by plugging it back into the context of the problem to see if it makes sense.

Example Problem

Problem: A chemist wants to create a 10% solution. She has a 25% solution and a 5% solution. How much of each solution should she mix to obtain 100 mL of the 10% solution?

Steps to Solve

1. Define Variables:

   • Let \( x \) = volume of the 25% solution (in mL).
   • Let \( y \) = volume of the 5% solution (in mL).

2. Set Up Proportions:

   • We know the total solution volume should be 100 mL: \[ x + y = 100 \]
   • We also want the final concentration to be 10%: \[ 0.25x + 0.05y = 0.10(100) \]

3. Write Equations: We already have the two equations:

   • Equation 1: \( x + y = 100 \)
   • Equation 2: \( 0.25x + 0.05y = 10 \)

4. Solve the Equations: From Equation 1, we can express \( y \): \[ y = 100 - x \] Substitute \( y \) in Equation 2: \[ 0.25x + 0.05(100 - x) = 10 \] Simplifying: \[ 0.25x + 5 - 0.05x = 10 \] \[ 0.20x + 5 = 10 \] \[ 0.20x = 5 \] \[ x = 25 \] Now, substitute \( x \) back to find \( y \): \[ y = 100 - 25 = 75 \]

5. Final Answer: You need 25 mL of the 25% solution and 75 mL of the 5% solution.

Quick Check

• Validate the total: \( 25 + 75 = 100 \) mL ✔️
• Validate the concentration:
  • Amount of solute from 25% solution: \( 0.25 \times 25 = 6.25 \) mL
  • Amount of solute from 5% solution: \( 0.05 \times 75 = 3.75 \) mL
  • Total solute: \( 6.25 + 3.75 = 10 \) mL.
  • Concentration: \( \frac{10}{100} = 0.10 \) or 10% ✔️

Thus, the solution is validated, and the calculations are correct.