Proportions: Direct versus Inverse

Ratios and proportions are commonly encountered in mathematics and chemistry.  Ratios compare two numbers.  At the grocery store, you will commonly find examples of ratios when looking at sale items:  buy 3 candy bars for $2. This is a ratio.

\frac{3 candy bars}{2 dollars}

A proportion is an equation relating two different ratios.  For example, you are at the grocery store and see the candy sale.  You want to buy $6 worth of candy bars.  How many do you end up buying on sale?  The answer is 9.  Let’s set up a proportion to solve this prolem.

 \frac{3 candy bars}{2 dollars} = \frac{x candy bars}{6 dollars}

As you can see in the equation above, the number of candy bars is on the top of the fraction and dollars is on the bottom.  We must set up proportions so that the type of quantities on the top and bottom of the fraction (the numerator and the denominator) are the same on both sides of the equation.  We put candy bars on top and dollars on the bottom.

What does it mean when we say “salary increases directly proportional with age?”  We can write this expression mathematically as follows;  A=S where A is for Age and S is for Salary.  This expression says that as age varies, the same changes will be observed in salary;  if age increases, salary also increases.  On the other hand if age decreases, salary will also decrease.  Another way to look at this relationship is using the Cartesian plane.  Since our relationship is directly proportional and as one quantity increases, the other quantity will also increase, our relationship can be visualized using a straight line with a positive slope.

Inversely proportional is opposite of directly proportional.  If salary was inversely proportional to age, we would expect salary to decrease as age increases.  Another way to write this relationship is as follows.  S=\frac{1}{A}.  Let’s try out this equation.  As Age increases, the fraction becomes smaller and smaller. If Age is 20, then Salary will be equal to 1/20 or 0.05. When age increases to 80, salary decreases to 1/80 or 0.0125. In the Cartesian plane, inversely proportional relationships are represented by a hyperbola.

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