Market concentration

Alligation is an old and practical method of solving arithmetic problems related to mixtures of ingredients. There are two types of alligation: alligation medial, used to find the quantity of a mixture given the quantities of its ingredients, and alligation alternate, used to find the amount of each ingredient needed to make a mixture of a given quantity. Alligation medial is merely a matter of finding a weighted mean. Alligation alternate is more complicated and involves organizing the ingredients into high and low pairs which are then traded off.

Examples

Alligation medial

Suppose you make a cocktail drink combination out of 1/2 Coke, 1/4 Sprite, and 1/4 orange soda. The Coke has 120 grams of sugar per liter, the Sprite has 100 grams of sugar per liter, and the orange soda has 150 grams of sugar per liter. How much sugar does the drink have? This is an example of alligation medial because you want to find the amount of sugar in the mixture given the amounts of sugar in its ingredients. The solution is just to find the weighted average by composition:

$\frac{1}{2}\times 120+\frac{1}{4}\times 100+\frac{1}{4}\times 150=122.5$ grams per liter

Alligation alternate

Suppose you like 1% milk, but you have only 3% whole milk and ½% low fat milk. How much of each should you mix to make an 8 ounce cup of 1% milk? This is an example of alligation alternate because you want to find the amount of two ingredients to mix to form a mixture with a given amount of fat. Since there are only two ingredients, there is only one possible way to form a pair. The difference of 3% from the desired 1%, 2%, is assigned to the low fat milk, and the difference of ½% from the desired 1%, ½%, is assigned alternately to the whole milk. The total amount, 8 ounces, is then divided by the sum $2+\frac{1}{2}=\frac{5}{2}$ to yield $\frac{16}{5}$, and the amounts of the two ingredients are

$\frac{16}{5}\times \frac{1}{2}=\frac{8}{5}$ ounces whole milk and $\frac{16}{5}\times 2=\frac{32}{5}$ ounces low fat milk.

A general formula that works for both alligation "alternate" and alligation "medial" is the following: Aa + Bb = Cc.

In this formula, A is the volume of ingredient A and a is its mixture coefficient (i.e. a= 3%); B is volume of ingredient B and b is its mixture coefficient; and C is the desired volume C, and c is its mixture coefficient. So in the above example we get: A(0.03) + B(0.005) = 8oz(0.01). We know B = (8oz-A), and so can easily solve for A and B to get 1.6 and 6.4oz, respectively. Using this formula you can solve for any of the 6 variables A,a,B,b,C,c, regardless of whether you're dealing with medial, alternate, etc.

References

Alligation, Forerunner of Linear Programming, Frederick V. Waugh, Journal of Farm Economics Vol. 40, No. 1 (Feb., 1958), pp. 89–103 http://www.jstor.org/stable/1235348

External links

• Alligation alterne et medial: www.formatp.ca/alligation.php
• Alligation Alternate and the Composition of Medicines: Arithmetic and Medicine in Early Modern England
• Robinson's Progressive Practical Arithmetic