How many quarts of pure antifreeze must be added to 3 quarts of a 10​% antifreeze solution to obtain a 20​% antifreeze​ solution?

Let's solve this problem step by step.

We are given:
- 3 quarts of a 10% antifreeze solution.
- We need to find how many quarts of pure antifreeze (100% antifreeze) must be added to obtain a 20% antifreeze solution.

Let's denote the amount of pure antifreeze to be added as \( x \) quarts.

### Step 1: Determine the amount of pure antifreeze in the original solution.
The original solution contains 10% antifreeze, so the amount of pure antifreeze in the original 3 quarts is:
\[
\text{Amount of pure antifreeze} = 0.10 \times 3 = 0.3 \text{ quarts}
\]

### Step 2: Set up the equation.
When \( x \) quarts of pure antifreeze is added, the total amount of antifreeze in the solution becomes \( 0.3 + x \) quarts. The total volume of the solution will then be \( 3 + x \) quarts.

We need the concentration of the antifreeze in this new solution to be 20%. Therefore, the equation is:
\[
\frac{0.3 + x}{3 + x} = 0.20
\]

### Step 3: Solve the equation.
Multiply both sides of the equation by \( 3 + x \) to eliminate the denominator:
\[
0.3 + x = 0.20(3 + x)
\]

Distribute the 0.20 on the right side:
\[
0.3 + x = 0.6 + 0.20x
\]

Subtract \( 0.20x \) from both sides to get all \( x \) terms on one side:
\[
x - 0.20x = 0.6 - 0.3
\]

Simplify the equation:
\[
0.80x = 0.3
\]

Divide both sides by 0.80 to solve for \( x \):
\[
x = \frac{0.3}{0.80} = 0.375 \text{ quarts}
\]

### Final Answer:
You need to add **0.375 quarts** of pure antifreeze to the 3 quarts of a 10% antifreeze solution to obtain a 20% antifreeze solution.

Would you like any further details or have any questions about this solution?

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Here are 5 related questions you might consider:
1. How would the solution change if you wanted a 30% antifreeze solution instead?
2. What if you had 5 quarts of a 10% antifreeze solution instead of 3 quarts?
3. How much pure water should be added to a 20% antifreeze solution to reduce it to a 10% solution?
4. If you added 1 quart of pure antifreeze, what would be the resulting concentration?
5. How would you calculate the amount of antifreeze in a solution if you know the total volume and concentration?

**Tip:** When solving mixture problems, always set up an equation that relates the amount of the substance of interest before and after the mixture to the desired concentration.

Learn how to calculate the concentration of antifreeze in a solution using a step-by-step guide. This problem involves determining how many quarts of pure antifreeze to add to a 3-quart 10% antifreeze solution to achieve a 20% antifreeze solution. Suitable for high school students and anyone interested in understanding mixture problems and percentages.

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