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College Mathematics Notes Section 2.7 Page 1 of 4

Chapter 2: Algebra Basics

Section 2.7: Direct and Inverse Variation Problems

Big Idea: Direct and inverse variations are fancy words for familiar concepts that lead to basic equations tosolve.

You know that when you drive for a longer time (while keeping your speed constant), then you end up going afurther distance, and that if you buy more stuff at the store, then you are going to pay more in sales tax. Thedistance traveled depends on the amount of time spent traveling, and the amount of sales tax paid depends onthe amount of the purchase. In addition, you know that this dependence is linear, or directlyproportional,in the sense that if you drive for twice the number of hours, then you expect to go twice as far, and if you buythree times as much stuff, then you expect to pay three times as much sales tax. This section deals with thesekinds of relationships.

If two quantities vary directly or are directly proportional, (i.e., we say, distance varies directly with time,or sales tax is directly proportional to purchase amount) then we write the equation describing the relationship

as:

the first quantity equals a constant times the second quantity

distance = speed time

sales tax = percentage purchase amount

or, in general, if y varies directly with x, or y is directly proportional to x, then:y = kx

where kis the constant of proportionality orrate of change.

Here are some examples of everyday proportionality constants:

1. Speed: miles per hour2. Sales Tax rate: $ of tax per $ of purchase3. Water flow: gallons per minute4. Gas: $ per gallon5. Oatmeal: $ per ounce6. Wood: $ per foot7. Monetary conversion: euros per dollar8. Anything where one quantity has a simple (linear) dependence on another quantity

To solve a direct variation problem:

1) Write down the direct variation equation: Quantity1 = constant Quantity2

2) Put given numbers in for Quantity1 and Quantity2, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.

Practice:

1. A mechanic earns $736.00 for 32 hours of work. What will the mechanic earn for 40 hours of work?

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2. On 10/20/2010, 10.00 euros was worth $13.95. How many euros is $10,000.00 worth?

3. The area of a sphere varies directly as the square of its radius. If a sphere with a radius of 2.50 metershas an area of 78.5 square meters, find the area of a sphere with a radius of 5.2 cm. Use the correctnumber of significant figures in your answer.

Joint variation is the case where one quantity is directly proportional to the product of two or more quantities.To solve a joint variation problem:

1) Write down the direct variation equation: Quantity1 = constant Quantity2 Quantity3

2) Put given numbers in for the quantities, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.

Practice:

4. P varies jointly with rand s. If P = 16 when r= 5 and s = -8, find P when r= 2 and s = 10.

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5. The value of a piece of real estate varies jointly with the size of the lot, the size of the house on the lot,and the distance of the lot from the sewage treatment plant. If a 0.50 acre plot with a 2,500 square foothouse that is 5.5 miles from the treatment plant is worth $285,000, find the value of a 1.0 acre plot witha 2,800 square foot house that is 1.75 miles from the plant.

Inverse variation, orinverse proportion is the case where one quantity is proportional to the reciprocal ofanother quantity. Examples include: your homework grade is inversely proportional to the number of hoursspent watching TV, or the loudness of a sound you hear is inversely proportional to the square of the distancefrom the source.

1 ky k y

x x= =

To solve an inverse variation problem:

1) Write down the inverse variation equation: Quantity1 = constant / Quantity22) Put given numbers in for the quantities, then solve for the constant.3) Use the constant in the direct variation equation to find the desired quantity.

Practice:

6. A student who watches 8 hours of TV per week has an average homework score of 94%. If thehomework score is inversely proportional to the number of hours of TV watched, find the average scoreof a student who watches 20 hours of TV per week.

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7. The resistance of a circuit element is inversely proportional to the current flowing through it. If thecurrent = 25 Amps for a resistance of 100. ohms, find the resistance when the current is 73.8 amps.

8. The pressure of a gas depends on the amount of gas present (the number of moles, n), the absolutetemperature (T), and the volume occupied by the gas (V). The relationship between these variables isthat the pressure of a gas varies as the product of the amount of gas and the absolute temperature andinversely as the volume occupied by the gas. IfPis 133,000 Pascals when Tis 373 K, n = 120.3 moles,

and Vis 2.8 cubic meters, what is Pwhen Tis 400. K, Vis 1.9 cubic meters, and n = 15.5 moles? Usethe correct number of significant figures in your answer.

9. A drive gear having 80 teeth and rotating at 40 rpm meshes with a second gear having 16 teeth. Howfast does the second gear rotate?

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