🧠 The theory: why C1V1 = C2V2 works
Strength multiplied by volume gives the amount of drug present. When you dilute, you add solvent but not drug, so the amount of drug before and after must be equal. That is exactly what the equation says: the strength times volume on the left (before) equals the strength times volume on the right (after). Because both sides are the same amount of drug, you can rearrange to find whichever value is missing, and the units cancel, so percentages can be used directly.
C1 x V1 = C2 x V2
Finding the volume of stock
Worked example
What volume of a 10% w/v stock is needed to make 200 mL of a 2% w/v solution?
- Step 1 — Put the known values into the equation. C1 is 10%, C2 is 2%, V2 is 200 mL, and V1 (the stock volume) is what you want: 10 x V1 = 2 x 200.
- Step 2 — Work out the right-hand side: 2 x 200 = 400.
- Step 3 — Divide by C1 to find V1: 400 / 10 = 40.
Answer: 40 mL of stock, then make up to 200 mL with diluent (which means adding 160 mL)
Worked example
How much of a 5% stock makes 500 mL of a 0.05% solution?
- Step 1 — Set up the equation: 5 x V1 = 0.05 x 500.
- Step 2 — Right-hand side: 0.05 x 500 = 25.
- Step 3 — Divide by 5: 25 / 5 = 5.
Answer: 5 mL of stock, made up to 500 mL
Harder: how much water to add
A question may ask for the volume of water to ADD rather than the final volume. Find the final volume with the equation, then subtract the starting volume.
Worked example
How much water must be added to 50 mL of a 20% solution to make it 5%?
- Step 1 — Here you know the starting strength and volume and the final strength, and you want the final volume. Set up 20 x 50 = 5 x V2.
- Step 2 — Left-hand side: 20 x 50 = 1000.
- Step 3 — Divide by the final strength to get V2: 1000 / 5 = 200 mL.
- Step 4 — The question asks for water to ADD, so subtract the starting volume: 200 minus 50 = 150.
Answer: 150 mL of water
Worked example
How much water is added to 100 mL of a 10% solution to make 4%?
- Step 1 — Find the final volume: 10 x 100 = 4 x V2, so V2 = 1000 / 4 = 250 mL.
- Step 2 — Subtract the starting volume: 250 minus 100 = 150.
Answer: 150 mL of water
Harder: dilution factors
Worked example
20 mL of a 15% solution is made up to a final volume of 60 mL. What is the final strength?
- Step 1 — Set up the equation with the final strength as the unknown: 15 x 20 = C2 x 60.
- Step 2 — Left-hand side: 15 x 20 = 300.
- Step 3 — Divide by the final volume: 300 / 60 = 5.
Answer: 5% w/v
Tip: Quick check: 60 / 20 = 3, a 1 in 3 dilution, and 15% / 3 = 5%.
⚠️ Add versus make up to
Read carefully. Make up to means bring the total to that final volume. Add means the extra volume on top of what you started with. The two answers differ by the starting volume, and the exam tests whether you noticed.
Test yourself
What volume of a 20% stock makes 500 mL of a 4% solution?
Answer: 20 x V1 = 4 x 500; V1 = 2000 / 20 = 100 mL
How much water is added to 200 mL of a 25% solution to make 5%?
Answer: Final volume = (25 x 200) / 5 = 1000 mL; water to add = 1000 - 200 = 800 mL
30 mL of a 10% solution is made up to 150 mL. What is the final strength?
Answer: 10 x 30 = C2 x 150; C2 = 300 / 150 = 2%
What volume of a 50% solution makes 200 mL of a 5% solution?
Answer: 50 x V1 = 5 x 200; V1 = 1000 / 50 = 20 mL