A doctor prescribes an ointment that is 4% hydrocortisone. A pharmacist has 1% and 5% concentrations in stock. How many ounces of each should the pharmacist use to make a 3-ounce tube?
What if it was 2% instead of 4%?
If x oz. of 1% are used, then the rest (3-x) are 5%, so
.01x + .05(3-x) = .04(3)
x = 3/4
So, 3/4 oz of 1% and 9/4 oz of 5%
Note that 4% is 3/4 of the way from 1% to 5%, so 3/4 of the total is the 5% strength.
To find out how many ounces of each concentration the pharmacist should use, we can set up a system of equations.
Let's assume x represents the amount of the 1% concentration in ounces, and y represents the amount of the 5% concentration in ounces.
Given that the total amount of the ointment needed is 3 ounces, we have the equation:
x + y = 3
Also, since the ointment is 4% hydrocortisone, we can set up another equation based on the concentration:
(0.01 * x) + (0.05 * y) = 0.04 * 3
Simplifying the second equation:
0.01x + 0.05y = 0.12
Now, we can solve the system of equations to find the values of x and y.
Using the substitution method, we can solve the first equation for x:
x = 3 - y
Substituting this value of x into the second equation:
0.01(3 - y) + 0.05y = 0.12
Expanding and simplifying the equation:
0.03 - 0.01y + 0.05y = 0.12
0.04y = 0.09
y = 0.09 / 0.04
y = 2.25
Now, substitute the value of y back into the first equation to find x:
x + 2.25 = 3
x = 3 - 2.25
x = 0.75
Therefore, the pharmacist should use 0.75 ounces of the 1% concentration and 2.25 ounces of the 5% concentration to make a 3-ounce tube of ointment.
To determine how many ounces of each concentration the pharmacist should use, we can set up an equation based on the given information.
Let's assume x represents the number of ounces of the 1% concentration ointment that the pharmacist should use and y represents the number of ounces of the 5% concentration ointment.
Based on the given information, the total amount of ointment is 3 ounces. Therefore, we can write the equation:
x + y = 3 ---(Equation 1)
Now let's consider the hydrocortisone content in the ointment. The doctor prescribes a 4% hydrocortisone ointment, which means the total amount of hydrocortisone in the final 3-ounce tube should be 0.04 * 3 = 0.12 ounces.
The 1% concentration ointment contains 0.01 ounces of hydrocortisone per ounce, so the hydrocortisone content in x ounces of the 1% ointment is 0.01 * x = 0.01x ounces.
Similarly, the 5% concentration ointment contains 0.05 ounces of hydrocortisone per ounce, so the hydrocortisone content in y ounces of the 5% ointment is 0.05 * y = 0.05y ounces.
Since we want to have a total of 0.12 ounces of hydrocortisone in the final mixture, we can set up another equation:
0.01x + 0.05y = 0.12 ---(Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of x and y.
Solving the system of equations, we get:
x + y = 3 (Equation 1)
0.01x + 0.05y = 0.12 (Equation 2)
To solve this system, we can use different methods such as substitution or elimination. Let's use the substitution method here.
From Equation 1, we can rewrite it as x = 3 - y and substitute this value for x in Equation 2:
0.01(3 - y) + 0.05y = 0.12
0.03 - 0.01y + 0.05y = 0.12
0.04y = 0.09
y = 0.09 / 0.04
y = 2.25
Now, substitute the value of y back into Equation 1 to find x:
x + 2.25 = 3
x = 3 - 2.25
x = 0.75
Therefore, the pharmacist should use 0.75 ounces of the 1% concentration ointment and 2.25 ounces of the 5% concentration ointment to make a 3-ounce tube of 4% hydrocortisone ointment.