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A proportion is a mathematical statement expressing equality of two ratios. [1] [2]: =: a and d are called extremes, b and c are called means. Proportion can be written as =, where ratios are expressed as fractions.
First, you have to understand the problem. [2] After understanding, make a plan. [3] Carry out the plan. [4] Look back on your work. [5] How could it be better? If this technique fails, Pólya advises: [6] "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
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.
When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.
The incorrect additive strategy can be expressed as L + R = 10; this is a constant sum relation. Here is the graph for these two equations. For the numeric values involved in the problem statement, these graphs are "similar" and it is easy to see why individuals consider their incorrect answers perfectly reasonable.
It is also observed that 1 gram of chlorine reacts with 1.19 g of iodine. This ratio of 1.19 obeys the law because it is a simple fraction (1/3) of 3.58. (This is because it corresponds to the formula ICl 3, which is one known compound of iodine and chlorine.) Similarly, hydrogen, carbon, and oxygen follow the law of reciprocal proportions.
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Pacioli's work likewise teaches through examples, but it also develops arguments for the validity of its solutions through reference to general principles, axioms and logical proof. In this way the Summa begins to reintegrate the logical methods of classical Greek geometry into the medieval discipline of algebra.
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