How many ways are there to adjust two quantities so that they are in a given proportional relationship? explain your reasoning.

There are two ways, direct proportionality and inverse proportionality.

In mathematics, proportionality is the relationship between two quantities where the ratio or product is constant and not zero. In the first case the relationship is directly proportional, in the second inversely proportional.  

Two quantities are directly proportional if one is a multiple of the other. For values not equal to zero, the ratio of the two quantities is then constant. The quantities go hand in hand. Their constant ratio is the proportionality constant or proportionality factor. If one quantity increases by a certain factor, the other quantity also increases by that same factor.  

Two quantities are inversely proportional if their product is constant and not zero. If one quantity increases by a certain factor, the other quantity decreases by that same factor.

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on monday rodney ate 1/4th of the sandwich = 3/12

on tuesday he ate 1/3rd of the sandwich = 4/12

therefore in total he has eaten = 3/12 + 4/12 = 7/12

thus we can say that he has eaten 7/12 of the sandwich on monday and tuesday alltogether.

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