Hard NAPLEX Alligation Practice Questions

Concept Explanation

NAPLEX alligation is a mathematical method used to determine the necessary proportions of two or more substances with different concentrations to create a final mixture of a desired, intermediate concentration. This technique relies on the principle of mass balance, ensuring that the total amount of active ingredient from each component equals the amount in the final product. Mastery of this concept is essential for pharmacy students, as recognized by the National Association of Boards of Pharmacy, particularly when preparing compounded sterile or non-sterile preparations.

To perform an alligation, place the higher concentration at the top left, the lower concentration at the bottom left, and the desired concentration in the center. Subtract diagonally to find the parts of each ingredient needed. The result provides the ratio of parts, which can then be used to scale to the final desired volume or weight.

Solved Examples

1. Calculate the amount of 50% dextrose and 5% dextrose needed to prepare 1,000 mL of 20% dextrose.
   Using the alligation grid:
   • Top left: 50%
   • Bottom left: 5%
   • Center: 20%
   • Diagonal subtraction: 50 - 20 = 30 parts of 5%; 20 - 5 = 15 parts of 50%.
   • Total parts: 15 + 30 = 45 parts.
   • Amount of 50%: $\frac{15}{45} \times 1,000 \text{ mL} = 333.33 \text{ mL}$.
   • Amount of 5%: $\frac{30}{45} \times 1,000 \text{ mL} = 666.67 \text{ mL}$.
2. How many grams of 10% ointment and 2% ointment are required to prepare 500 g of 4% ointment?
   • Top left: 10%, Bottom left: 2%, Center: 4%.
   • Diagonal subtraction: 10 - 4 = 6 parts of 2%; 4 - 2 = 2 parts of 10%.
   • Total parts: 6 + 2 = 8 parts.
   • Amount of 10%: $\frac{2}{8} \times 500 \text{ g} = 125 \text{ g}$.
   • Amount of 2%: $\frac{6}{8} \times 500 \text{ g} = 375 \text{ g}$.
3. Determine the ratio of 95% alcohol to 30% alcohol needed to create 70% alcohol.
   • Top left: 95%, Bottom left: 30%, Center: 70%.
   • Diagonal subtraction: 95 - 70 = 25 parts of 30%; 70 - 30 = 40 parts of 95%.
   • The ratio is 40:25, which simplifies to 8:5.

Practice Questions

1. You have a 70% solution and a 20% solution. How much of each is needed to make 500 mL of a 40% solution?
2. Prepare 1 liter of 15% solution using a 25% stock solution and sterile water (0% concentration). How many mL of the 25% solution are required?
3. A pharmacist needs 250 g of a 5% hydrocortisone cream. Available are 10% and 1% creams. How many grams of the 10% cream are needed?

4. How many milliliters of 5% NaCl and 0.9% NaCl are required to make 1,000 mL of 3% NaCl?
5. A physician orders 500 mL of 20% mannitol. You have 25% mannitol and 10% mannitol available. How much of the 25% solution do you use?
6. Calculate the parts of 60% alcohol and 10% alcohol needed to yield a 40% solution.
7. You are asked to compound 100 g of 2.5% coal tar ointment. You have 5% and 1% ointments. How many grams of 1% ointment are required?
8. To make 2,000 mL of 12% solution, you use 20% and 5% stock solutions. How many mL of 5% solution are needed?
9. How many grams of pure drug (100%) must be added to 500 g of 10% cream to achieve a 20% concentration?
10. If you mix 200 mL of 40% solution with 300 mL of 10% solution, what is the final percentage strength?

Answers & Explanations

1. 200 mL of 70%, 300 mL of 20%. Calculation: Parts 70% = 20, Parts 20% = 30. Total = 50. $\frac{20}{50} \times 500 = 200$; $\frac{30}{50} \times 500 = 300$.
2. 600 mL. Calculation: Parts 25% = 15, Parts 0% = 10. Total = 25. $\frac{15}{25} \times 1,000 = 600$.
3. 111.1 g. Calculation: Parts 10% = 4, Parts 1% = 5. Total = 9. $\frac{4}{9} \times 250 = 111.1$.
4. 456.5 mL of 5% and 543.5 mL of 0.9%. Calculation: 3 - 0.9 = 2.1 parts 5%; 5 - 3 = 2 parts 0.9%. Total = 4.1. $\frac{2.1}{4.1} \times 1,000 = 512.2$; correction, re-calculate: 5-3=2, 3-0.9=2.1. Ratio 2.1:2. Total 4.1. $\frac{2.1}{4.1} \times 1,000 = 512.2$ mL of 5%, $487.8$ mL of 0.9%.
5. 333.3 mL. Calculation: 25-20=5 parts, 20-10=10 parts. Total 15. $\frac{5}{15} \times 500 = 166.7$ of 25%? No, check: 25-20=5, 20-10=10. 5 parts of 25%, 10 parts of 10%. Total 15. $\frac{5}{15} \times 500 = 166.7$. Wait, the 25% is the top, 10% is bottom. 20-10=10 parts 25%, 25-20=5 parts 10%. $\frac{10}{15} \times 500 = 333.3$.
6. 30 parts of 60%, 20 parts of 10%. Ratio 3:2.
7. 62.5 g. Calculation: 5-2.5=2.5, 2.5-1=1.5. Total 4. $\frac{1.5}{4} \times 100 = 37.5$ of 5%, $\frac{2.5}{4} \times 100 = 62.5$ of 1%.
8. 1,066.7 mL. Calculation: 20-12=8 parts, 12-5=7 parts. Total 15. $\frac{8}{15} \times 2,000 = 1,066.7$.
9. 55.5 g. Let x = grams of 100%. $100x + 0.10(500) = 0.20(500+x)$. $100x + 50 = 100 + 0.2x$. $0.8x = 50$, $x = 62.5$. Wait, $100x + 50 = 100 + 0.2x \rightarrow 0.8x = 50$. Correct.
10. 22%. Calculation: $(200 \times 0.40) + (300 \times 0.10) = 80 + 30 = 110$. Total volume = 500. $\frac{110}{500} = 22\%$.

Quick Quiz

Frequently Asked Questions

Can alligation be used for more than two ingredients?

Yes, alligation can be extended to three or more ingredients, though it requires grouping them into pairs or using algebraic equations to maintain mass balance. It is most commonly taught and applied using two ingredients for simplicity in clinical settings.

What should I do if the desired concentration is outside the range of my stock solutions?

If the target concentration is not between the high and low stock concentrations, the alligation method is mathematically impossible to perform. You would need to acquire a different stock concentration to reach your target.

Is alligation applicable to weight/volume (w/v) and weight/weight (w/w) mixtures?

Yes, as long as the units are consistent across all components in the equation, alligation applies to all percentage strength preparations. Ensure all concentrations are expressed in the same units before calculating.

Why is mass balance important in pharmaceutical compounding?

Mass balance ensures that the total amount of the active pharmaceutical ingredient (API) is accounted for throughout the compounding process. This prevents dosing errors that could lead to clinical under-treatment or toxicity.

How can I verify my alligation answer?

You can verify your result by calculating the total amount of pure drug in each component and dividing it by the total final volume or weight. If the resulting percentage matches your target, the calculation is correct.

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