To determine how many grams of hydrocortisone 2.5% must be mixed with hydrocortisone 1% to prepare 30 grams of hydrocortisone 1.5% ointment, we can proceed through the following steps:
1. Let's denote the amount of 2.5% hydrocortisone ointment by [tex]\( x \)[/tex] grams.
2. Since the total amount of ointment needed is 30 grams, the amount of 1% hydrocortisone ointment will be [tex]\( 30 - x \)[/tex] grams.
3. Now, we need to set up an equation based on the percentages of hydrocortisone to ensure the final mixture has the desired concentration of 1.5%.
4. The amount of hydrocortisone in the 2.5% ointment is [tex]\( 0.025 \times x \)[/tex].
5. The amount of hydrocortisone in the 1% ointment is [tex]\( 0.01 \times (30 - x) \)[/tex].
6. The total amount of hydrocortisone in the final mixture should be [tex]\( 0.015 \times 30 \)[/tex] grams.
7. Therefore, we can write the equation:
[tex]\[
0.025x + 0.01(30 - x) = 0.015 \times 30
\][/tex]
8. Simplify the equation step by step:
[tex]\[
0.025x + 0.01 \times 30 - 0.01 \times x = 0.015 \times 30
\][/tex]
[tex]\[
0.025x + 0.3 - 0.01x = 0.45
\][/tex]
[tex]\[
0.015x + 0.3 = 0.45
\][/tex]
[tex]\[
0.015x = 0.15
\][/tex]
[tex]\[
x = \frac{0.15}{0.015}
\][/tex]
[tex]\[
x = 10
\][/tex]
So, the amount of 2.5% hydrocortisone ointment needed is [tex]\( 10 \)[/tex] grams.
To verify:
- If [tex]\( x = 10 \)[/tex] grams of 2.5% ointment, then [tex]\( 30 - 10 = 20 \)[/tex] grams of 1% ointment is used.
- The amount of hydrocortisone from the 2.5% ointment is [tex]\( 0.025 \times 10 = 0.25 \)[/tex] grams.
- The amount of hydrocortisone from the 1% ointment is [tex]\( 0.01 \times 20 = 0.2 \)[/tex] grams.
- The total hydrocortisone in the mixture is [tex]\( 0.25 + 0.2 = 0.45 \)[/tex] grams.
- The total mass is 30 grams, so the concentration is [tex]\( \frac{0.45}{30} = 0.015 \)[/tex] or 1.5%.
Hence, the correct answer is:
O B. 10 gm
