Question
solve mixture priblems whith proportions quick check
Answers
GPT-4o mini
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.
### 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.