Question
Solve mixture problems with proportions quick check connections academy
Answers
GPT-4o mini
Sure! Mixture problems typically involve finding the concentration or amount of substances in a mixture based on given proportions. Here’s a general approach to solving these types of problems, along with a quick example to illustrate.
### Steps to Solve Mixture Problems:
1. **Identify the substances involved**: Determine what is being mixed and their respective amounts/concentrations.
2. **Set up a proportion or equation**: Use the concept of ratios and proportions to set up an equation based on the information given.
3. **Solve for the unknown**: Isolate the variable you need to find and solve the equation.
4. **Check your answer**: Ensure that the solution fits into the context of the problem and makes sense logically.
### Example Problem:
**Problem**: You have a 10% salt solution and a 30% salt solution. If you mix 4 liters of the 10% solution with \(x\) liters of the 30% solution, how much of the 30% solution do you need to make a 20% salt solution?
**Solution Steps**:
1. **Identify the components**:
- Amount of 10% solution = 4 liters.
- Concentration of 10% solution = 0.10.
- Amount of 30% solution = \(x\) liters.
- Concentration of 30% solution = 0.30.
- Desired concentration after mixing = 20%.
2. **Set up the equation**: The total amount of salt in the mix equals the total concentration of the salt in the resulting mixture.
\[
\text{Salt from 10% solution} + \text{Salt from 30% solution} = \text{Salt in resulting solution}
\]
This can be expressed as:
\[
(0.10)(4) + (0.30)(x) = (0.20)(4 + x)
\]
3. **Simplify and solve for \(x\)**:
\[
0.4 + 0.30x = 0.8 + 0.20x
\]
Rearranging gives:
\[
0.30x - 0.20x = 0.8 - 0.4
\]
\[
0.10x = 0.4
\]
Solving for \(x\):
\[
x = \frac{0.4}{0.10} = 4
\]
4. **Conclusion**: You need to mix **4 liters** of the 30% solution with the 4 liters of the 10% solution to achieve a 20% salt solution.
### General Tips:
- Always double-check your calculations.
- Ensure the units are consistent.
- Pay attention to the final concentration, as it must fit logically within the context of the problem.
If you have a specific mixture problem you'd like help with, feel free to share!
### Steps to Solve Mixture Problems:
1. **Identify the substances involved**: Determine what is being mixed and their respective amounts/concentrations.
2. **Set up a proportion or equation**: Use the concept of ratios and proportions to set up an equation based on the information given.
3. **Solve for the unknown**: Isolate the variable you need to find and solve the equation.
4. **Check your answer**: Ensure that the solution fits into the context of the problem and makes sense logically.
### Example Problem:
**Problem**: You have a 10% salt solution and a 30% salt solution. If you mix 4 liters of the 10% solution with \(x\) liters of the 30% solution, how much of the 30% solution do you need to make a 20% salt solution?
**Solution Steps**:
1. **Identify the components**:
- Amount of 10% solution = 4 liters.
- Concentration of 10% solution = 0.10.
- Amount of 30% solution = \(x\) liters.
- Concentration of 30% solution = 0.30.
- Desired concentration after mixing = 20%.
2. **Set up the equation**: The total amount of salt in the mix equals the total concentration of the salt in the resulting mixture.
\[
\text{Salt from 10% solution} + \text{Salt from 30% solution} = \text{Salt in resulting solution}
\]
This can be expressed as:
\[
(0.10)(4) + (0.30)(x) = (0.20)(4 + x)
\]
3. **Simplify and solve for \(x\)**:
\[
0.4 + 0.30x = 0.8 + 0.20x
\]
Rearranging gives:
\[
0.30x - 0.20x = 0.8 - 0.4
\]
\[
0.10x = 0.4
\]
Solving for \(x\):
\[
x = \frac{0.4}{0.10} = 4
\]
4. **Conclusion**: You need to mix **4 liters** of the 30% solution with the 4 liters of the 10% solution to achieve a 20% salt solution.
### General Tips:
- Always double-check your calculations.
- Ensure the units are consistent.
- Pay attention to the final concentration, as it must fit logically within the context of the problem.
If you have a specific mixture problem you'd like help with, feel free to share!