To solve this problem, we need to set up a system of equations and solve it to find out how many liters of each type of juice should be mixed. Let's break it down step-by-step:
1. Define the Variables:
- Let \(\( x \)\) be the amount of the 15% orange juice (in liters).
- Let \(\( y \)\) be the amount of the 9% orange juice (in liters).
2. Set Up the Equations:
- The total volume of the mixture should be 30 liters. Therefore, we can write:
\(\[
x + y = 30
\]\)
- The mixture should be 13% orange juice. The total amount of orange juice contributed by both types of juice should be 13% of 30 liters. Therefore, the second equation is:
\(\[
0.15x + 0.09y = 0.13 \times 30
\]\)
3. Simplify the Second Equation:
- First, calculate \(\( 0.13 \times 30 \)\):
\(\[
0.13 \times 30 = 3.9
\]\)
- So, our second equation becomes:
\(\[
0.15x + 0.09y = 3.9
\]\)
Now we have the system of equations:
\(\[
\begin{cases}
x + y = 30 \\
0.15x + 0.09y = 3.9
\end{cases}
\]\)
4. Solve the System of Equations:
Substitution Method:
- Solve the first equation for \(\( y \)\):
\(\[
y = 30 - x
\]\)
- Substitute \(\( y \)\) in the second equation:
\(\[
0.15x + 0.09(30 - x) = 3.9
\]\)
- Simplify and solve for \(\( x \)\):
\(\[
0.15x + 2.7 - 0.09x = 3.9
\]\)
\(\[
0.06x + 2.7 = 3.9
\]\)
\(\[
0.06x = 1.2
\]\)
\(\[
x = \frac{1.2}{0.06} = 20
\]\)
- Now, substitute \(\( x = 20 \)\) back into the first equation to find \(\( y \)\):
\(\[
y = 30 - 20 = 10
\]\)
So, the solution to the system is:
- \(\( x = 20 \)\) liters of the 15% orange juice
- \(\( y = 10 \)\) liters of the 9% orange juice
Therefore, you should mix 20 liters of the 15% orange juice with 10 liters of the 9% orange juice to get 30 liters of a mixture that is 13% orange juice.
