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The amount of active ingredient does not change when you dilute. That gives the relationship: initial concentration times initial volume equals final concentration times final volume.
One idea, expressed as one equation, applied to most of the questions in this area.
The principle
Diluting adds solvent. It does not add or remove active ingredient. So the amount of active is the same before and after.
Write that as an equation and you get the initial concentration times the initial volume equalling the final concentration times the final volume.
Worked
You have 70 percent stock. You need 500 mL of 35 percent.
Rearranged, the volume of stock needed is the final concentration times the final volume, divided by the initial concentration. Thirty-five times 500 divided by 70 is 250 mL.
Then add diluent to make it up to 500 mL, which means 250 mL of diluent.
Those are different numbers and the stem will ask for one of them specifically. Here it is 250 mL of stock and 250 mL of diluent, which happen to be equal - a coincidence of this example that will not hold generally. Read which one is wanted before you answer.
Units must match
Concentrations in the same form on both sides, volumes in the same unit on both sides. Mixing percentage with mg per mL, or mL with liters, is the error that produces an answer out by a factor of a thousand.
Converting everything to mg per mL first costs a few seconds and removes the risk entirely.
Concentrating rather than diluting
The same equation works in reverse when solvent is evaporated or a stronger product is added. The invariant is still the amount of active.
Where alligation comes in
Dilution handles a stock and a pure diluent. Alligation handles two preparations that both contain active.
A diluent at 0 percent is just the special case where one of the two strengths is zero, which is why the same worked example can be done either way.
Common questions
What is the dilution equation?
Initial concentration times initial volume equals final concentration times final volume.
Why does it work?
Diluting does not change the amount of active ingredient, only the volume it sits in.
What is the common trap?
Giving the volume of stock when the question asked for the volume of diluent to add, or the reverse.
What about units?
They must match on both sides. Converting everything to mg per mL first removes the risk.
When do I use alligation instead?
When both preparations contain active. Dilution is the special case where one is a pure diluent.