Module 4 of 14 · 12 min
Alligation
Whenever you mix two strengths of the same thing — two creams, two dextrose bags, alcohol of two strengths, or a cream plus pure powder — alligation gives the proportions in one quick grid. It is a favourite compounding question on both exams.
🧠 Cross the diagonals: each ingredient's parts come from the OTHER corner. Closer to d → more of it.
Alligation alternate answers 'how much of each?' when you mix a stronger (H) and a weaker (L) preparation to get a desired strength (d). Draw a grid: H top-left, L bottom-left, d in the middle. Subtract along the diagonals (ignore signs): the stronger gets $d - L$ parts, the weaker gets $H - d$ parts.
Pure drug counts as 100%; a base, water or petrolatum with no drug counts as 0%. If one ingredient's amount is already fixed (e.g. 'add to 90 g of 2% ointment'), match that amount to its parts and scale the other — don't scale against the total.
Alligation medial answers the opposite question: 'what strength do I get?' Weighted average: $\%_{\text{final}} = \frac{\sum(\text{amount}\times\%)}{\sum\text{amount}}$.
When one of the two components is 0% (pure diluent), alligation and $C_1V_1 = C_2V_2$ give the same answer — use whichever is quicker. Alligation is genuinely needed when BOTH components contain drug.
🧭 The method
- 1Check the desired strength lies between the two you have (otherwise it can't be made).
- 2Grid: H and L on the left, d in the centre; parts of H = d − L, parts of L = H − d.
- 3Total parts = sum. If the total amount is given: each amount = total × (its parts / total parts).
- 4If one component's amount is fixed instead: amount of other = fixed amount × (other parts / fixed parts).
- 5Check with alligation medial: the mixture's % should come back to d.
Before each reveal, predict the next step — then check.
How many grams of 10% and 1% ointment should be mixed to make 120 g of 3% ointment?
- 1
Set up the grid and subtract along the diagonals.
$$\begin{array}{ccc} 10\% & & 3-1 = 2\ \text{parts}\\ & 3\% & \\ 1\% & & 10-3 = 7\ \text{parts}\end{array}$$ Total parts.
$$2 + 7 = 9\ \text{parts} \equiv 120\,\text{g}$$Share the 120 g in the ratio of the parts.
$$10\%: \ 120\,\text{g}\times\frac{2}{9} = 26.67\,\text{g}\qquad 1\%: \ 120\,\text{g}\times\frac{7}{9} = 93.33\,\text{g}$$Check with alligation medial: drug in = drug wanted.
$$26.67\times 0.10 + 93.33\times 0.01 = 2.667 + 0.933 = 3.6\,\text{g} = 3\%\ \text{of}\ 120\,\text{g}\ \checkmark$$
Same method — the last steps are yours. Work on paper, then reveal.
How many grams of hydrocortisone powder (100%) and of 1% hydrocortisone cream make 60 g of 2.5% cream?
- 1
Pure powder is 100%. Grid.
$$\begin{array}{ccc} 100\% & & 2.5-1 = 1.5\ \text{parts}\\ & 2.5\% & \\ 1\% & & 100-2.5 = 97.5\ \text{parts}\end{array}$$ - 2
Total parts.
$$1.5 + 97.5 = 99\ \text{parts}\equiv 60\,\text{g}$$ - 3
Your turn
Now you: grams of each.
Work it out on paper, then reveal.
$$\text{powder} = 60\times\frac{1.5}{99} = 0.91\,\text{g}\qquad\text{1\% cream} = 60\times\frac{97.5}{99} = 59.09\,\text{g}$$ - 4
Your turn
Now you: check that the mixture contains 2.5% of 60 g.
Work it out on paper, then reveal.
$$0.909\times 1 + 59.09\times 0.01 = 0.909 + 0.591 = 1.5\,\text{g} = 2.5\%\ \text{of}\ 60\,\text{g}\ \checkmark$$
Type your number. Your first try decides when this module comes back for review.
How many mL of dextrose 20% must be mixed with dextrose 5% to make 500 mL of dextrose 10%?
Parts of 20% = 10 − 5; parts of 5% = 20 − 10.
= 166.67 mL
What is the final % w/v when 200 mL of 40%, 300 mL of 25% and 500 mL of 10% are mixed?
Alligation medial: weight each strength by its volume — don't just average the three percentages.
= 20.5 %
How many grams of white petrolatum (0%) must be added to 30 g of 5% ointment to reduce it to 2%?
Petrolatum = 0%. The 30 g of ointment is fixed — match it to its parts.
= 45 g
How many grams of salicylic acid powder (100%) must be added to 90 g of 2% salicylic acid ointment to make it 5%?
Not 3% of 90 g: the powder you add also increases the total weight. Use alligation with 100%.
= 2.84 g
How many mL of 95% v/v ethanol and 40% v/v ethanol are needed to make 1000 mL of 70% v/v? Enter the mL of 95%.
Parts of 95% = 70 − 40; parts of 40% = 95 − 70. (Volumes contract on mixing — in practice make up to volume.)
= 545.45 mL
⚠️ Tricky
How many grams of 0.5% hydrocortisone cream must be mixed with 40 g of 2.5% hydrocortisone cream to make a 1% cream?
The 40 g is fixed — match it to the stronger cream's parts, not to the total.
= 120 g
⚠️ The trap: Treating 40 g as the TOTAL and taking 1.5/2 of it gives 30 g.
⚠️ Tricky
How many mL of a 1:1000 solution must be mixed with a 1:5000 solution to make 500 mL of a 1:2500 solution?
Alligation needs strengths, and a bigger ratio number is a WEAKER solution. Convert each to %.
= 125 mL
⚠️ The trap: Alligating the ratio numbers 1000, 5000 and 2500 directly reverses the parts and gives 312.5 mL.
⚠️ Tricky
Dextrose 50% and dextrose 5% are mixed to make 1 L of dextrose 12.5%. How many mL of the 5% solution are needed?
Work out both parts — the question asks for the weaker one.
= 833.33 mL
⚠️ The trap: Reporting the stronger component's volume gives 166.67 mL.
⚠️ Tricky
0.5 kg of 4% ointment, 250 g of 1% ointment and 250 g of plain base are mixed. What is the final % w/w?
Same units for every amount, and the plain base (0%) still adds weight to the total.
= 2.25 % w/w
⚠️ The trap: Leaving the drug-free base out of the denominator gives 3%.
⚠️ Tricky
A 1 L parenteral nutrition bag must supply 680 kcal from dextrose, made from dextrose 70% and dextrose 10%. Dextrose monohydrate (as in IV solutions) provides 3.4 kcal/g; carbohydrate in general is often quoted as 4 kcal/g. How many mL of dextrose 70% are needed?
Calories → grams of dextrose → % in 1 L → alligation. Use the factor for IV dextrose.
= 166.67 mL
⚠️ The trap: Using 4 kcal/g gives 170 g (17%) and only 116.67 mL of dextrose 70%.
🕵️ Spot the mistake
Make 100 g of 4% cream from 10% and 2% creams. How much 10% cream?
A student wrote
⚠️ Traps examiners set
- Putting the parts on the wrong ingredient — the stronger gets (d − L), not (H − d).
- Forgetting that pure drug is 100% and plain base is 0%.
- Scaling against the total when one component's amount is fixed.
- Strengthening by adding (d − L)% of the starting weight — the added powder increases the total.
- Averaging percentages in alligation medial without weighting by amount.
💬 Explain it in your own words
Why does the ingredient whose strength is FARTHER from the desired strength get FEWER parts? Explain it as a balance (see-saw) around the desired percentage.