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The concentration model

As well as the three common fraction models, there is a fourth model, the concentration model, which can be useful in addressing the intensive nature of fractions. Fractions can be described as intensive quantities because they represent the relationship between quantities rather than the quantities themselves. Mixing one cup of soda water with two cups of orange juice produces three cups of drink in total, with orange juice two-thirds of the mix. However, if you mix two cups of soda water with four cups of orange juice you will have six cups of drink in total although the fraction of orange juice in the mix (an intensive quantity) remains the same.

The concentration model

Thinking about fractions using the concentration model helps to appreciate their nature as intensive quantities. The concentration model can also provide simple answers to what otherwise might appear to be difficult fraction question. In particular, if you add 1 to both the numerator and denominator of a fraction does it become, larger, smaller or stay the same? As a specific example using fraction notation, we may ask is 131/250 larger, smaller or the same as 132/251?

An analogy (the concentration model) will help solve this problem. Suppose you added 131 millilitres of orange juice to 119 millilitres of soda water. What fraction of the mixture is orange juice? If I add another millilitre of orange juice, the mixture clearly has more orange juice. What fraction of the new mixture is orange juice? Remember that the total orange soda mixtureis now 251 millilitres.

It now is much easier to determine which is larger (or stronger), 131/250 or 132/251? 

More generally, using the concentration model, adding the same number to both the top and bottom of a fraction makes the fraction larger. Adding changes the concentration, multiplying the top and bottom is needed to keep the concentration the same. This is another way of showing what it means to say that fractions represent a multiplicative structure not an additive structure.