United States
Hong KongAlligation
Alligation finds the proportions of two preparations needed to produce an intermediate strength. Subtract diagonally: each part is the difference between the other strength and the target.
The calculation with the worst reputation on this exam, and it is two subtractions.
What it is for
You have a stronger preparation and a weaker one. You need something in between. Alligation tells you the ratio to mix them in.
The target must sit between the two strengths. If it does not, no mixture of them will reach it, and that is occasionally the point of the question.
The method
- Write the higher strength top left, the lower strength bottom left, the target in the middle
- Subtract diagonally, always taking the smaller from the larger
- The result on each line is the number of parts of the strength on that line
- Add the parts to get the total, then apportion the final volume
Worked
You have 70 percent and you have a diluent at 0 percent. You need 500 mL at 35 percent.
Target minus lower is 35 minus 0, which is 35 parts of the 70 percent. Higher minus target is 70 minus 35, which is 35 parts of the diluent.
Thirty-five parts each, so 70 parts total, so half and half. Two hundred and fifty mL of each.
Thirty-five is exactly halfway between 0 and 70, so an equal mixture is obviously right. Whenever the target sits at the midpoint the answer is one to one, and if your grid says otherwise you have subtracted in the wrong direction. Use the easy cases to verify your method rather than to practice it.
When the numbers are not tidy
The method does not change. Target minus lower gives the parts of the higher; higher minus target gives the parts of the lower. Then apportion.
The diagonal is the only thing to remember, and remembering it as "each strength gets the difference on the far side" survives better than remembering a picture.
The alternative
You can do any alligation as a pair of simultaneous equations, or as a mass balance: the amount of active in the final mixture equals the amount contributed by each component.
That is slower but it never goes wrong, and it is a good check when a grid answer looks implausible.
Common questions
What is alligation used for?
Finding the proportions of two preparations needed to produce an intermediate strength.
How does the method work?
Subtract diagonally. Each strength gets the difference between the target and the other strength.
What if the target is halfway between?
The answer is one to one. Use that as a check on your method.
What if the target is outside both strengths?
No mixture will reach it, and that is sometimes what the question is testing.
Is there an alternative method?
Yes, a mass balance. Slower, but it never goes wrong and makes a good check.