Paula Scott, a massage therapist needs 3 ounces of a 20% lavender
oil solution. She has only 5% and 30% lavender oil solutions available.
How many ounces of each should Paula mix to obtain the desired
solution?
To determine how many ounces of each solution Paula should mix, we need to set up an equation based on the amount of lavender oil in each solution. Let's assume she needs x ounces of the 5% lavender oil solution and y ounces of the 30% lavender oil solution.
The equation can be set up as follows:
0.05x + 0.30y = 0.20 * 3
This equation represents the total amount of lavender oil (in ounces) in the desired mixture. The left side of the equation calculates the amount of lavender oil in the 5% and 30% solutions, while the right side represents the total amount of lavender oil needed in the final 20% solution (which is 3 ounces multiplied by 20%).
Now, let's solve this equation to find the values of x and y.
0.05x + 0.30y = 0.6
To make the equation easier to solve, let's multiply through by 100 to eliminate the decimals:
5x + 30y = 60
We now have a system of linear equations. However, we still need another equation to solve for both x and y. One constraint in this problem is that Paula needs a total of 3 ounces of the final mixture:
x + y = 3
Now we have a system of two equations:
5x + 30y = 60
x + y = 3
There are different methods to solve this system of equations, such as substitution or elimination. Let's use the substitution method here.
We can solve the second equation for x:
x = 3 - y
Substitute this value of x into the first equation:
5(3 - y) + 30y = 60
15 - 5y + 30y = 60
25y = 45
y = 45/25
y = 1.8
Now that we have the value of y, we can substitute it back into the second equation to find x:
x + 1.8 = 3
x = 3 - 1.8
x = 1.2
Therefore, Paula should mix 1.2 ounces of the 5% lavender oil solution with 1.8 ounces of the 30% lavender oil solution to obtain the desired 3-ounce solution containing 20% lavender oil.