How many quarts of pure antifreeze must be added 2 quarts of a 10% antifreeze solution to obtain a 30% antifreeze solution

To determine how many quarts of pure antifreeze must be added to 2 quarts of a 10% antifreeze solution to obtain a 30% antifreeze solution, we can set up an algebraic equation.

### Step 1: Define the variables.
Let \( x \) be the number of quarts of pure antifreeze to be added.

### Step 2: Calculate the amount of pure antifreeze in the original solution.
- The original solution is 2 quarts of a 10% antifreeze solution.
- The amount of pure antifreeze in this solution is \( 0.1 \times 2 = 0.2 \) quarts.

### Step 3: Set up the equation.
- After adding \( x \) quarts of pure antifreeze, the total volume of the solution will be \( 2 + x \) quarts.
- The amount of pure antifreeze in the new solution is \( 0.2 + x \) quarts.
- We want the final solution to be 30% antifreeze:
  \[
  \frac{0.2 + x}{2 + x} = 0.3
  \]

### Step 4: Solve the equation.
1. Multiply both sides by \( 2 + x \) to clear the fraction:
   \[
   0.2 + x = 0.3(2 + x)
   \]
2. Distribute the 0.3 on the right side:
   \[
   0.2 + x = 0.6 + 0.3x
   \]
3. Subtract \( 0.3x \) from both sides:
   \[
   0.7x = 0.4
   \]
4. Divide both sides by 0.7:
   \[
   x = \frac{0.4}{0.7} \approx 0.571
   \]

### Final Answer:
You must add approximately **0.571 quarts** of pure antifreeze to achieve a 30% antifreeze solution.

Would you like more details on this process, or do you have any questions?

Here are some related problems to explore:
1. How much pure antifreeze is needed to increase a 5% solution to 20%?
2. What is the concentration if 0.5 quarts of pure antifreeze are added to the original solution?
3. How much 50% antifreeze solution should be added to get a 30% concentration?
4. What is the new concentration if 1 quart of water is added instead?
5. How many quarts of 100% antifreeze should be added to a 15% solution to make it 50%?

**Tip:** When working with mixture problems, remember that setting up the ratio correctly is key to finding the solution efficiently.

Learn how to calculate the amount of pure antifreeze needed to achieve a 30% antifreeze solution by adding to a 10% solution. This problem involves mixture problems, percentage calculations, and algebraic equations. Suitable for high school students.

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