Answer :
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 [tex]\( x \)[/tex] be the amount of the 15% orange juice (in liters).
- Let [tex]\( y \)[/tex] 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:
[tex]\[ x + y = 30 \][/tex]
- 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:
[tex]\[ 0.15x + 0.09y = 0.13 \times 30 \][/tex]
3. Simplify the Second Equation:
- First, calculate [tex]\( 0.13 \times 30 \)[/tex]:
[tex]\[ 0.13 \times 30 = 3.9 \][/tex]
- So, our second equation becomes:
[tex]\[ 0.15x + 0.09y = 3.9 \][/tex]
Now we have the system of equations:
[tex]\[ \begin{cases} x + y = 30 \\ 0.15x + 0.09y = 3.9 \end{cases} \][/tex]
4. Solve the System of Equations:
Substitution Method:
- Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = 30 - x \][/tex]
- Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 0.15x + 0.09(30 - x) = 3.9 \][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 0.15x + 2.7 - 0.09x = 3.9 \][/tex]
[tex]\[ 0.06x + 2.7 = 3.9 \][/tex]
[tex]\[ 0.06x = 1.2 \][/tex]
[tex]\[ x = \frac{1.2}{0.06} = 20 \][/tex]
- Now, substitute [tex]\( x = 20 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 30 - 20 = 10 \][/tex]
So, the solution to the system is:
- [tex]\( x = 20 \)[/tex] liters of the 15% orange juice
- [tex]\( y = 10 \)[/tex] 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.
1. Define the Variables:
- Let [tex]\( x \)[/tex] be the amount of the 15% orange juice (in liters).
- Let [tex]\( y \)[/tex] 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:
[tex]\[ x + y = 30 \][/tex]
- 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:
[tex]\[ 0.15x + 0.09y = 0.13 \times 30 \][/tex]
3. Simplify the Second Equation:
- First, calculate [tex]\( 0.13 \times 30 \)[/tex]:
[tex]\[ 0.13 \times 30 = 3.9 \][/tex]
- So, our second equation becomes:
[tex]\[ 0.15x + 0.09y = 3.9 \][/tex]
Now we have the system of equations:
[tex]\[ \begin{cases} x + y = 30 \\ 0.15x + 0.09y = 3.9 \end{cases} \][/tex]
4. Solve the System of Equations:
Substitution Method:
- Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y = 30 - x \][/tex]
- Substitute [tex]\( y \)[/tex] in the second equation:
[tex]\[ 0.15x + 0.09(30 - x) = 3.9 \][/tex]
- Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 0.15x + 2.7 - 0.09x = 3.9 \][/tex]
[tex]\[ 0.06x + 2.7 = 3.9 \][/tex]
[tex]\[ 0.06x = 1.2 \][/tex]
[tex]\[ x = \frac{1.2}{0.06} = 20 \][/tex]
- Now, substitute [tex]\( x = 20 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 30 - 20 = 10 \][/tex]
So, the solution to the system is:
- [tex]\( x = 20 \)[/tex] liters of the 15% orange juice
- [tex]\( y = 10 \)[/tex] 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.
