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newSchoolWorks.com › VoTechWorks › Health & Personal Services › Pharmacy Technician › Pharmacy Calculations and Order Entry › Lesson 8
HLT-PHAR41-03 · Lesson 8 · 15 clock hours · theory

Dilutions and alligation

Objective. Solve stock-dilution (C1V1 = C2V2) and alligation problems for liquids and semisolids.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by a licensed pharmacist or by a certification or approval body. It is a study draft and may contain errors. Do not use it to make decisions about any patient or medication. Always follow your instructor, your supervising professional, and current official references and regulations.

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Why Dilutions and Alligation Matter

Pharmacy technicians routinely prepare products that are not commercially available at the exact strength a prescriber orders. Two mathematical tools handle the vast majority of these situations: the stock-dilution formula and alligation. Mastering both is required for accurate compounding and order entry, and errors in either can cause serious patient harm.

The Stock-Dilution Formula: C₁V₁ = C₂V₂

When a concentrated (stock) solution is mixed with a diluent—usually purified water, normal saline, or another compatible vehicle—the amount of active ingredient stays the same; only the volume changes. This relationship is captured by:

> C₁V₁ = C₂V₂

Where:

Concentration can be expressed as a percentage (w/v, v/v), ratio strength, or mg/mL—as long as both C values use the same units. Likewise, both V values must share a unit (mL, L, etc.).

Steps to Solve a Dilution Problem

1. Identify the four variables; three will be given, one unknown.

2. Confirm units match on each side.

3. Rearrange algebraically to isolate the unknown.

4. Calculate and label the answer.

5. Verify: the final concentration must be lower than the starting concentration (you are diluting, not concentrating).

Volume of Diluent

After finding V₁ (stock volume needed), subtract it from V₂ to find how much diluent to add:

> Volume of diluent = V₂ − V₁

Alligation: Mixing Two Strengths

Alligation is used when no single stock solution matches the desired concentration. Instead, you blend a higher-strength product with a lower-strength product (which may be a diluent at 0%).

Setting Up the Alligation Grid

Draw a tic-tac-toe grid:

```

Higher strength (H) | | Parts of lower strength = D − L

| Desired |

Lower strength (L) | (D) | Parts of higher strength = H − D

```

Fill in:

Then subtract diagonally:

The two diagonal differences give the ratio of parts.

Converting Parts to Volumes

1. Add the parts: total parts = parts of H + parts of L.

2. Divide each component's parts by total parts to get its fraction.

3. Multiply each fraction by the total volume needed.

Alligation for Semisolids

The same grid applies to ointments, creams, and gels. Volume is simply replaced by weight (grams). For example, mixing a 10% zinc oxide ointment with plain ointment base (0%) to make a 4% product uses the identical grid method.

Important Reminders

Key terms8

C₁V₁ = C₂V₂The stock-dilution equation stating that the quantity of solute before dilution equals the quantity after dilution; concentration times volume on each side must be equal.
Stock solutionA concentrated solution that is diluted to produce a weaker, ready-to-use preparation.
DiluentAn inert liquid (such as purified water or normal saline) added to a stock solution to reduce its concentration.
AlligationA method of calculating the proportions of two preparations of different strengths needed to produce a mixture of an intermediate desired strength.
Desired concentration (D)In alligation, the target strength of the finished preparation, placed in the center of the alligation grid.
PartsThe relative proportions of each component determined by alligation; they represent a ratio, not a volume or weight, until converted using the total quantity needed.
Percent weight-in-volume (% w/v)A concentration expression meaning grams of solute per 100 mL of solution; for example, a 5% w/v solution contains 5 g per 100 mL.
Compounding recordA document that captures the formula, quantities, lot numbers, and technician information for each compounded preparation, required for quality assurance and traceability.

Worked examples4

1. How many mL of a 70% isopropyl alcohol stock solution are needed to prepare 500 mL of a 30% isopropyl alcohol solution? How much purified water must be added?
  1. Identify variables: C₁ = 70%, V₁ = ?, C₂ = 30%, V₂ = 500 mL.
  2. Apply C₁V₁ = C₂V₂: 70 × V₁ = 30 × 500.
  3. Solve for V₁: V₁ = 15,000 ÷ 70 = 214.29 mL of stock solution.
  4. Volume of diluent = V₂ − V₁ = 500 − 214.29 = 285.71 mL of purified water.
Answer: 214.29 mL of 70% isopropyl alcohol + 285.71 mL of purified water
2. A prescriber orders 240 mL of a 0.025% (w/v) solution. The pharmacy stock is a 0.1% (w/v) solution. How many mL of stock solution are needed?
  1. Identify variables: C₁ = 0.1%, V₁ = ?, C₂ = 0.025%, V₂ = 240 mL.
  2. Apply C₁V₁ = C₂V₂: 0.1 × V₁ = 0.025 × 240.
  3. Solve: V₁ = 6 ÷ 0.1 = 60 mL of stock solution.
  4. Volume of diluent = 240 − 60 = 180 mL.
Answer: 60 mL of the 0.1% stock solution
3. Using alligation, calculate how many mL of a 95% ethanol and how many mL of purified water (0%) are needed to prepare 1,000 mL of 70% ethanol.
  1. Set up the alligation grid: H = 95, D = 70, L = 0.
  2. Diagonal differences: parts of H = D − L = 70 − 0 = 70; parts of L = H − D = 95 − 70 = 25.
  3. Total parts = 70 + 25 = 95.
  4. Volume of 95% ethanol = (70 ÷ 95) × 1,000 = 736.84 mL.
  5. Volume of purified water = (25 ÷ 95) × 1,000 = 263.16 mL.
  6. Verify: 736.84 + 263.16 = 1,000 mL. ✓
Answer: 736.84 mL of 95% ethanol + 263.16 mL of purified water
4. A compounding order calls for 60 g of a 6% zinc oxide ointment. The pharmacy has 10% zinc oxide ointment and plain ointment base (0%). How many grams of each are needed?
  1. Set up the alligation grid: H = 10%, D = 6%, L = 0%.
  2. Parts of 10% ointment = D − L = 6 − 0 = 6 parts.
  3. Parts of plain base = H − D = 10 − 6 = 4 parts.
  4. Total parts = 6 + 4 = 10.
  5. Grams of 10% ointment = (6 ÷ 10) × 60 = 36 g.
  6. Grams of plain base = (4 ÷ 10) × 60 = 24 g.
  7. Verify: 36 + 24 = 60 g. ✓
Answer: 36 g of 10% zinc oxide ointment + 24 g of plain ointment base

The math in this lesson2

Learners rearrange C1V1 = C2V2 to find 214.29 mL of 70% alcohol for 500 mL of 30%, plus alligation.
7.RP.3A-CED.4
Brush up in the math halls at newSchoolWorks.com:
In class: Dilution problem sets · Alligation grid practice · Compounding-record preview
Materials: Calculation workbook
Check: Quiz ≥ 90%

Practice check16

Answer each question, then check it. Nothing is recorded; this is your own practice.

Order Entry and Processing
1. How many mL of 70% stock are needed to make 200 mL of a 35% solution?
Order Entry and Processing
2. Alligation: mix 10% and 2% to make 400 g of 5%. How many grams of the 10% product are needed?
Order Entry and Processing
3. How many mL of 95% stock are needed to make 500 mL of a 70% solution?
Order Entry and Processing
4. Alligation: mix 20% and 5% to make 300 g of 10%. How many grams of the 20% product are needed?
Order Entry and Processing
5. How many mL of 20% stock are needed to make 100 mL of a 5% solution?
Order Entry and Processing
6. How many mL of 10% stock are needed to make 250 mL of a 2% solution?
Order Entry and Processing
7. How many mL of 50% stock are needed to make 500 mL of a 10% solution?
Order Entry and Processing
8. Alligation: mix 50% and 10% to make 200 g of 20%. How many grams of the 50% product are needed?
Order Entry and Processing
9. Which formula correctly represents the relationship used in stock-solution dilution problems?
Order Entry and Processing
10. A technician needs to prepare 200 mL of a 2% solution from a 10% stock solution. How many mL of the stock solution are required?
Order Entry and Processing
11. After calculating the volume of stock solution needed (V₁), how does a technician determine the volume of diluent to add?
Order Entry and Processing
12. In an alligation grid, the number placed in the CENTER of the grid represents which value?
Order Entry and Processing
13. A technician uses alligation to mix a 20% cream with a 5% cream to make a 12% cream. How many parts of the 20% cream are needed?
Order Entry and Processing
14. Using alligation with a 20% cream and a 5% cream to prepare a 12% cream, how many grams of the 5% cream are needed to make a total of 120 g of the 12% product?
Patient Safety and Quality Assurance
15. Which of the following is the PRIMARY purpose of a compounding record when preparing a diluted or alligated product?
Patient Safety and Quality Assurance
16. A technician is asked to prepare 150 mL of a 0.5% solution using a 2.5% stock solution. After calculating the volume of stock needed, the technician notices the result is larger than 150 mL. What does this most likely indicate?
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