Alligations are used to figure out how much of a higher & lower percent strength of the same drug should be combined to make a desired amount of the drug at a new percent strength. This new strength must be in between the two original strengths for alligations to work. We calculate those amounts, or parts, needed of each drug/active ingredient for the final preparation by using a grid system & some basic math.
A Percent of lower strength as a whole number  D Difference of desired & higher strength (absolute value) 

C Percent of desired strength as a whole number  
B Percent of higher strength as a whole number  E Difference of desired & lower strength (absolute value) 
We find the answers by taking the following steps:
 Take the difference of the desired & lower strength. This represents the parts of the higher strength needed.
Absolute value of (A – C) = E  Take the difference of the desired & higher strength. This represents the parts of the lower strength needed.
Absolute value of (B – C) = D 
Now write each of these individual differences as a fraction over the sum of both differences.
E / (D + E) represents the fraction of the
higher strength needed.D / (D + E) represents the fraction of the
lower strength needed.  Now that the individual parts are written as fractions, you can just multiply each one by the final quantity of the desired preparation to figure out the quantities needed of the lower & higher strengths!
Let's watch a video illustrating this technique. For your convenience, this video is looped & does not have any loud background music nor intros & outros. You may have to press play on mobile/tablet devices. Consider the following problem: We have in stock 70% & 40% isopropyl alcohol. We need 300 mL of 60% alcohol. How much of each do we mix together? The answer is (20/30) * 300 mL = 200 mL of the 70% alcohol & (10/30) * 300 = 100 mL of the 40% alcohol.