How many gallons each of 30​% alcohol and 10​% alcohol should be mixed to obtain 20 gal of 14​% ​alcohol?

We can solve this problem using a system of linear equations. Let:

- \( x \) be the number of gallons of 30% alcohol.
- \( y \) be the number of gallons of 10% alcohol.

We are given two important pieces of information:

1. The total amount of alcohol solution after mixing is 20 gallons.
2. The final mixture should be 14% alcohol.

### Step 1: Total amount of solution
We know the total volume of the mixture is 20 gallons:
\[
x + y = 20
\]

### Step 2: Total amount of alcohol in the solution
The amount of alcohol in each solution is calculated as:
- \( 0.30x \) gallons of pure alcohol from the 30% solution,
- \( 0.10y \) gallons of pure alcohol from the 10% solution.

The total amount of pure alcohol in the final 20-gallon mixture must be:
\[
0.14 \times 20 = 2.8 \, \text{gallons}.
\]

So, the equation for the amount of alcohol is:
\[
0.30x + 0.10y = 2.8
\]

### Step 3: Solving the system of equations
We now have two equations:
1. \( x + y = 20 \)
2. \( 0.30x + 0.10y = 2.8 \)

We can solve these equations simultaneously.

#### From the first equation, solve for \( y \):
\[
y = 20 - x
\]

#### Substitute this into the second equation:
\[
0.30x + 0.10(20 - x) = 2.8
\]
\[
0.30x + 2 - 0.10x = 2.8
\]
\[
0.20x + 2 = 2.8
\]
\[
0.20x = 0.8
\]
\[
x = \frac{0.8}{0.20} = 4
\]

#### Now, substitute \( x = 4 \) into the equation \( y = 20 - x \):
\[
y = 20 - 4 = 16
\]

### Conclusion
You should mix **4 gallons of 30% alcohol** with **16 gallons of 10% alcohol** to obtain 20 gallons of a 14% alcohol solution.

Would you like a breakdown of each step or have any questions?

Here are 5 related questions for practice:
1. How many gallons of 40% alcohol and 20% alcohol are needed to make 30 gallons of 28% alcohol?
2. If you mix 10 gallons of 50% alcohol with water, what will be the final concentration of the solution?
3. How much 15% salt solution should be mixed with 25% salt solution to get 50 gallons of 20% salt solution?
4. What amount of 12% acid and 8% acid is needed to create 100 liters of 10% acid solution?
5. How can you adjust the concentration of a solution if you only have one type of alcohol?

**Tip:** Always check your final answer by substituting the values into the original conditions!

How many gallons each of 30% alcohol and 10% alcohol should be mixed to obtain 20 gal of 14% alcohol?

Learn how to solve a mixture problem involving 30% and 10% alcohol solutions to create 20 gallons of 14% alcohol. This step-by-step solution involves solving a system of linear equations and applying percentage concepts. Suitable for students in Grades 9-10.

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