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To solve the problem where a 60% antifreeze solution must be mixed with 100 gallons of a 15% antifreeze solution to achieve a final mixture that is 50% antifreeze, we can apply a system of equations. Let's break it down step by step.

### Step-by-Step Solution

1. **Define the variables**:
   - Let \( x \) be the number of gallons of the 60% antifreeze solution to be added.
   - The amount of antifreeze in the 100 gallons of 15% solution is \( 100 \times 0.15 = 15 \) gallons.
   - The amount of antifreeze in the \( x \) gallons of 60% solution is \( 0.60x \) gallons.

2. **Equation for total amount of antifreeze**:
   The total amount of antifreeze in the mixture should be equal to 50% of the total volume of the mixture. The total volume of the mixture is \( x + 100 \) gallons, and 50% of that is \( 0.50(x + 100) \).

3. **Set up the equation**:
   \[
   0.60x + 15 = 0.50(x + 100)
   \]

4. **Solve the equation**:
   Simplify the right-hand side:
   \[
   0.60x + 15 = 0.50x + 50
   \]
   Subtract \( 0.50x \) from both sides:
   \[
   0.10x + 15 = 50
   \]
   Subtract 15 from both sides:
   \[
   0.10x = 35
   \]
   Divide by 0.10:
   \[
   x = \frac{35}{0.10} = 350
   \]

So, you need **350 gallons** of the 60% antifreeze solution to mix with the 100 gallons of 15% antifreeze to get a 50% solution.

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Would you like more details or have any questions on the process?

### Relative questions:
1. How would the answer change if the concentration of the first solution were 70% instead of 60%?
2. What if you needed a final solution of 40% antifreeze instead of 50%?
3. How can we verify the final concentration of the solution?
4. How does the volume of the second solution affect the calculation?
5. Could you apply this method to mixing other types of chemical solutions?

### Tip:
Always verify your solution by plugging the values back into the equation to ensure the concentrations match.

How many gallons of a 60% antifreeze solution must be mixed with 100 gallons of 15% antifreeze to get a mixture that is 50% antifreeze?

Learn how to mix a 60% antifreeze solution with a 15% solution to obtain a 50% mixture. This step-by-step solution involves algebraic methods and is suitable for students in Grades 9-11. Find out how to calculate the required volumes.

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