🧠 The theory: it is a weighted average in reverse
Mixing two strengths gives a weighted average that lands between them. Alligation works that backwards: the closer the target is to one strength, the more of that strength you need. The parts method captures this with a simple rule, because the distance from the target to each strength sets how much of the OTHER strength you use. That is why parts of the stronger come from (desired minus weaker), and parts of the weaker from (stronger minus desired).
parts of the stronger = desired minus weaker
parts of the weaker = stronger minus desired
Worked example
In what ratio should a 12% and a 2% ointment be mixed to make a 6% ointment?
- Step 1 — Parts of the stronger (12%) come from desired minus weaker: 6 minus 2 = 4.
- Step 2 — Parts of the weaker (2%) come from stronger minus desired: 12 minus 6 = 6.
- Step 3 — Write the ratio and simplify: 4 to 6 = 2 to 3.
Answer: 2 parts of 12% to 3 parts of 2%
Tip: Check by weighted average: (2 x 12 + 3 x 2) / 5 = 30 / 5 = 6%.
Worked example
In what ratio should a 50% and a 20% solution be mixed to make 30%?
- Step 1 — Parts of 50% = 30 minus 20 = 10.
- Step 2 — Parts of 20% = 50 minus 30 = 20.
- Step 3 — Ratio 10 to 20 simplifies to 1 to 2.
Answer: 1 part of 50% to 2 parts of 20%
Scaling to a quantity
Worked example
Using a 2 to 3 ratio of 12% to 2%, how much of each makes 500 mL?
- Step 1 — Add the parts to get the total: 2 + 3 = 5 parts.
- Step 2 — Work out what one part is worth: 500 / 5 = 100 mL.
- Step 3 — Multiply each share: 12% needs 2 x 100 = 200 mL; 2% needs 3 x 100 = 300 mL.
Answer: 200 mL of 12% and 300 mL of 2%
Harder: when one quantity is fixed
Worked example
You have 200 mL of a 12% ointment. How much 2% ointment must be added to make 6%?
- Step 1 — Find the ratio as before: 12% to 2% is 2 to 3.
- Step 2 — The 200 mL of 12% represents the 2 parts, so one part is 200 / 2 = 100 mL.
- Step 3 — The 2% is 3 parts, so 3 x 100 = 300 mL.
Answer: 300 mL of the 2% ointment
Harder: mixing with a diluent
Worked example
In what ratio should a 10% cream be mixed with a diluent base to make a 4% cream?
- Step 1 — A plain base counts as 0%.
- Step 2 — Parts of 10% = 4 minus 0 = 4.
- Step 3 — Parts of base (0%) = 10 minus 4 = 6.
- Step 4 — Ratio 4 to 6 simplifies to 2 to 3.
Answer: 2 parts of 10% cream to 3 parts of base
💡 Sanity check
The desired strength must sit between the two starting strengths. If it does not, the mix is impossible and you have misread the question.
Test yourself
In what ratio should a 5% and a 1% cream be mixed to make a 2% cream?
Answer: Parts of 5% = 2 - 1 = 1; parts of 1% = 5 - 2 = 3; ratio 1 to 3 (5% to 1%)
In what ratio should a 20% and a 5% ointment be mixed to make 10%?
Answer: Parts of 20% = 10 - 5 = 5; parts of 5% = 20 - 10 = 10; ratio 1 to 2 (20% to 5%)
In what ratio should pure drug (100%) be mixed with base (0%) to make 20%?
Answer: Parts of 100% = 20 - 0 = 20; parts of 0% = 100 - 20 = 80; ratio 20 to 80 = 1 to 4
Using a 2 to 3 ratio of 12% to 2%, how much of each makes 250 g?
Answer: 5 parts = 250 g, so 1 part = 50 g; 100 g of 12% and 150 g of 2%