How much water should be added to 1 gallon of pure antifreeze to obtain a solution that is 10 % ​antifreeze?

To solve this problem, we can use the concept of concentration and the relationship between the amount of antifreeze and the total volume of the solution.

Let \( x \) be the amount of water (in gallons) that we need to add.

### Step 1: Set up the equation
- Initially, we have 1 gallon of pure antifreeze, which is 100% antifreeze.
- After adding \( x \) gallons of water, the total volume of the solution becomes \( 1 + x \) gallons.
- We want the final solution to be 10% antifreeze.

The amount of antifreeze in the solution remains 1 gallon, but the total volume is now \( 1 + x \) gallons.

The concentration of antifreeze in the solution can be expressed as:

\[
\text{Concentration of Antifreeze} = \frac{\text{Amount of Antifreeze}}{\text{Total Volume of Solution}} = \frac{1}{1 + x}
\]

We want this concentration to be 10%, or \( 0.10 \). Therefore, we set up the equation:

\[
\frac{1}{1 + x} = 0.10
\]

### Step 2: Solve the equation
Now, solve for \( x \):

\[
1 = 0.10 \times (1 + x)
\]

\[
1 = 0.10 + 0.10x
\]

\[
1 - 0.10 = 0.10x
\]

\[
0.90 = 0.10x
\]

\[
x = \frac{0.90}{0.10} = 9
\]

### Answer:
You need to add **9 gallons** of water to the 1 gallon of pure antifreeze to obtain a solution that is 10% antifreeze.

Would you like more details on this solution, or do you have any questions? Here are five related questions to explore further:

1. How would the amount of water change if the target concentration was 20% antifreeze?
2. What if you started with 2 gallons of pure antifreeze and wanted a 10% solution?
3. How much water would you add to a 50% antifreeze solution to dilute it to 10%?
4. How do you calculate the final concentration if a different volume of water is added?
5. Can you derive a general formula for the amount of water needed for any initial concentration?

**Tip:** Remember that when diluting a solution, the amount of solute remains the same, but the total volume changes, which affects the concentration.

Learn how to calculate the amount of water needed to dilute 1 gallon of pure antifreeze to obtain a 10% antifreeze solution. Step-by-step solution with concentration formula. Suitable for high school students studying chemistry or mathematics.

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