A doctor prescribes an ointment that is 2% hydrocortisone. A pharmacist has 1% and 5% concentrations in stock. How many ounces of each should the pharmacist use to make a 5-ounce tube?

Question

A doctor prescribes an ointment that is 2% hydrocortisone. A pharmacist has 1% and 5% concentrations in stock. How many ounces of each should the pharmacist use to make a 5-ounce tube?

Answer 1

to make 5 ounces of .02 from x ounces of 1% and y ounces of 5%

y=5-x

2-1 = 1

5-2 = 3

3/1 = 3 = y/x = (5-x)/x

3x = 5-x

4x = 5

x = 1.25 ounces of 1%

y = 3x = 3.75 ounces of 5%

check the solutions

.01(1.25) + .05(3.75)

= .0125 + .1875

= .1

= .02(5) = .1

to make 5 ounces of 2%

or

a more conventional method of solving the problem

.01x + .05(5-x) = .02(5)

.01x -.05x = .1 - .25

-.04x = -.15

x= -.15/.-04 = 1.25 ounces of 1%

y = 5-x = 5-1.25 = 3.75 ounces of 5%

Answer 2

We want to end up with 5 ounces of ointment that contains 2% hydrocortisone (and 98% jelly).

So we want to end up with 5 ounces of ointment that contains 1/10 of an ounce of hydrocortisone, because 2% of 5 ounces is 1/10 of an ounce.

Let x be the number of ounces of 1% hydrocortisone we will mix in (the "first ingredient").

x is the only variable we need.

Then 5-x will be the number of ounces of 5% hydrocortisone we will mix in (the "second ingredient"), because we want to end up with a total of 5 ounces of product.

x(1/100) is the number of ounces of cortisone in the first ingredient, because it is 1% cortisone.

(5-x)(5/100) is the number of ounces of cortisone in the second ingredient, because it is 5% cortisone.

The problem requires that

x(1/100) + (5-x)(5/100) = 1/10

Solving for x,

x/100 + (25 - 5x)/100 = 10/100

(25 - 4x)/100 = 10/100

25 - 4x = 10

x=15/4 = 3.75

So the answer is 3.75 ounces of the first ingredient (the 1% cortisone), and 1.25 ounces of the second ingredient (the 5% cortisone). It makes sense that we're putting in much more of the first ingredient than the second, because the desired concentration of the final product (2%) is closer to the concentration of the first ingredient (1%) than the second (5%).

Let's check the answer. The 3.75 ounces of the first ingredient contain .0375 ounces of cortisone.

The 1.25 ounces of the second ingredient contain .0625 ounces of cortisone.

.0375 + .0625 = .1 ounces, which is 2% of the 5 ounces of the final product.