16.2 Concentrations, Dilutions & Alligation

15.2 Concentrations, Dilutions & Alligation

Quick Answer: Concentration calculations tell you how much drug is in a given volume or mass, and dilution/alligation calculations tell you how to mix stocks to a target strength. Fluent conversions between % w/v, % w/w, mg/mL, and ratio strengths, plus C1V1 = C2V2 and alligation, are core Standard 3.4 skills. On FIB items, enter the exact numeric result (mL of stock, final volume, parts, or mg/mL) in the unit the stem requests.

Compounding and many clinical checks fail when “5%” is treated as a vague adjective rather than a defined quantity.

Percentage strengths

Weight in volume (% w/v)

% w/v = grams of solute in 100 mL of solution (used for solids dissolved in liquids).

• 5% w/v = 5 g in 100 mL
• In 50 mL of 5% w/v there are 2.5 g of solute

Weight in weight (% w/w)

% w/w = grams of solute in 100 g of preparation (common for ointments and some powders).

• 2% w/w hydrocortisone cream ≈ 2 g drug in 100 g cream
• For 30 g of 2% w/w: drug mass = 0.02 × 30 = 0.6 g = 600 mg

Do not casually swap w/v and w/w: density is not always 1 g/mL for thick bases.

Volume in volume (% v/v)

% v/v = mL of liquid solute per 100 mL of solution (common for ethanol/water mixtures).

Bridge to mg/mL (high-yield)

For dilute aqueous solutions, a practical identity is:

1% w/v = 1 g/100 mL = 1000 mg/100 mL = 10 mg/mL

Worked example A — % to mg/mL

Lidocaine 2% w/v as mg/mL:

• 2% = 2 × 10 = 20 mg/mL

Dose volume for 40 mg: 40 ÷ 20 = 2 mL.

Worked example B — mass in a given volume

How many milligrams of drug in 15 mL of 0.9% w/v sodium chloride?

• 0.9% w/v = 9 mg/mL
• 15 × 9 = 135 mg

(Alternatively: 0.9 g/100 mL → 0.135 g in 15 mL = 135 mg.)

Dilution formula: C1V1 = C2V2

When you dilute a solution without adding more drug, amount of solute is constant:

$C_1 V_1 = C_2 V_2$

• C1, V1 = concentration and volume of stock used
• C2, V2 = concentration and volume of final product

Rearrangements:

• V1 = (C2 × V2) / C1 → volume of stock needed
• V2 = (C1 × V1) / C2 → final volume after diluting a fixed stock aliquot
• Diluent to add (by volume, ideal mixing) ≈ V2 − V1

Concentrations must be in the same units (both %, both mg/mL, etc.).

Worked example C — find final volume

You dilute 20 mL of 10% w/v to 2% w/v. What is V2?

• 10 × 20 = 2 × V2
• V2 = 200 / 2 = 100 mL
• Diluent to add ≈ 100 − 20 = 80 mL

FIB if asked for final volume: 100 (mL). If asked for diluent: 80 (mL).

Worked example D — stock volume for a batch

Prepare 200 mL of 5% from 20% stock.

• V1 = (5 × 200) / 20 = 1000 / 20 = 50 mL of 20% stock
• Diluent ≈ 200 − 50 = 150 mL

Worked example E — mg/mL form

Stock 50 mg/mL, need 10 mg/mL in a final syringe volume of 5 mL.

• V1 = (10 × 5) / 50 = 1 mL stock
• Diluent = 4 mL (if final volume is 5 mL)

Alligation (mixing two strengths)

Alligation alternate finds the relative parts of a higher-strength and lower-strength product to obtain an intermediate strength.

Write:

     Higher % (H)
                    \  parts of H = (T − L)
         Target % (T)
                    /  parts of L = (H − T)
     Lower % (L)

Worked example F — classic alligation

Mix 10% and 2% to obtain 5%.

• Parts of 10% = 5 − 2 = 3
• Parts of 2% = 10 − 5 = 5
• Ratio 10% : 2% = 3 : 5

To make 80 mL of 5%:

• Total parts = 3 + 5 = 8
• Volume of 10% = (3/8) × 80 = 30 mL
• Volume of 2% = (5/8) × 80 = 50 mL

Check: drug from 10% = 0.10 × 30 = 3 g; from 2% = 0.02 × 50 = 1 g; total 4 g in 80 mL = 5% w/v. Good.

Worked example G — diluting with water (0%)

Mix 20% stock with water (0%) to make 5%.

• Parts of 20% = 5 − 0 = 5
• Parts of water = 20 − 5 = 15
• Ratio stock : water = 5 : 15 = 1 : 3

For 100 mL of 5%: stock = 25 mL, water = 75 mL (same result as C1V1: V1 = (5 × 100)/20 = 25 mL).

Alligation and C1V1 agree when one “strength” is zero; alligation shines when both components contain drug.

Ratio strengths

A ratio strength 1 : n (weight in volume) means 1 g of solute in n mL of solution.

Adrenaline (epinephrine) 1 : 1000 (classic emergency presentation):

• 1 g in 1000 mL = 1000 mg in 1000 mL = 1 mg/mL
• As % w/v: 1 g/1000 mL = 0.1 g/100 mL = 0.1% w/v

1 : 10 000 = 0.1 mg/mL = 0.01% w/v.

Worked example H — volume for a stated mass

How many mL of adrenaline 1 : 1000 provide 0.5 mg?

• Concentration = 1 mg/mL
• Volume = 0.5 ÷ 1 = 0.5 mL

FIB: 0.5 (mL). Decimal accuracy is critical—entering 5 would be a ten-fold overdose pattern.

Ethanol and other liquid dilutions

Treat stated % v/v ethanol with the same conservation idea when diluting pure or concentrated alcohol with water to a target % v/v (ideal volumes are approximate in real mixing because of volume contraction, but exam arithmetic usually uses C1V1 = C2V2).

Worked example I — ethanol

Prepare 200 mL of 30% v/v ethanol from 95% v/v stock.

• V1 = (30 × 200) / 95 = 6000 / 95 ≈ 63.16 mL of 95% ethanol
• Make up to 200 mL with water (as directed by the formula method in the stem)

If the exam wants a rounded value, follow the stated precision (for example 63.2 mL to one decimal place). When practising FIB, calculate fully then apply the required rounding once at the end.

Worked example J — parts from a concentrated disinfectant concept

Need 1 L of 2% from 10% concentrate:

• V1 = (2 × 1000) / 10 = 200 mL concentrate
• Diluent = 800 mL (approximately, for volume makeup)

FIB technique for concentration items

• Confirm whether the answer is stock volume, final volume, diluent volume, or concentration.
• Keep % and ratio conversions explicit: write “1% = 10 mg/mL” on scrap paper every time until automatic.
• For alligation, compute parts first, then scale to batch size—do not scale before parts.
• Sanity-check: final strength must lie between the two strengths mixed; final volume must be larger than stock volume when diluting down.

Concentrations, dilutions, and alligation convert labels into measurable recipes. Combined with Section 16.1 dosing, they let you move from “what strength is this?” to “how much do I measure?” without unit chaos—exactly what Standard 3.4 calculation items probe.