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Math Review of Direct, Inverse, Joint, and Combined Variation

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Equations that describe the relationship between two variables in a sentence express the variation between those variables.  Examples of types of variation include direct, inverse, joint, and combined variation.

What Is Direct Variation?

In direct variation, as one variable is multiplied by a constant and increases, another variable (the quotient) also increases.  Suppose Jane works at McDonald’s and is paid $9.35 per hour.  Her total wages vary directly with the amount of hours she works.  If she works 1 hour, her gross wages (before taxes) are $9.35, and if she works 2 hours her gross wages are $18.70.  If she works 8 hours, her wages are $74.80.  If the hours she works are plotted on the x axis against her wages on the y axis, the resulting line is a straight line showing direct variation.

What Is Inverse Variation?

Inverse variation is a relationship between variables so that as one variable decreases the other variable increases.  The equations expressing inverse variation take the form xy = k, where k is a constant, as well as y = k/x..  For example, the current c varies inversely with the resistance in ohms r.  When the current is 40 amps, the resistance is 12 ohms.  When the current is 24 amps, the resistance is 20 ohms.  If the current is plotted on the x axis, and the resistance is plotted on the y axis, the graph is in 2 separate curves called a hyperbola.

What Is Joint Variation?

Joint variation is a relationship between three variables, where one variable varies directly as the product of two or more variables.  The relationship between distance, rate, and time in motion problems is a good example of joint variation.  Suppose the rate is 60 miles per hour, and the time is 2 hours.  The distance is 120 miles.  Similarly, if the rate is 70 miles an hour and the time is 3 hours, the distance is 120 miles.

What Is Combined Variation?

Joint variation is a more complex relationship between three variables, where one variable varies directly as one variable and inversely as another.  The equations expressing combined variation take the form x = ky/z.  The force of attraction F of a body varies directly as its mass m times a constant k and inversely as the square of the distance d between the bodies.  The equation is F = km/d2, so if F equals 100 Newtons, m equals 8kg, and d equals 5 meters, then the equation is 100 = 8k/25.  Suppose the mass is 10 kg and the distance is 15 meters.  Then F will equal 13 8/9 Newtons.

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