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 to
solve.
You know that when you drive for a longer time (while keeping your speed constant), then you end up going a
further distance, and that if you buy more stuff at the store, then you are going to pay more in sales tax. The
distance traveled depends on the amount of time spent traveling, and the amount of sales tax paid depends on
the amount of the purchase. In addition, you know that this dependence is “linear”, or “directly proportional”,
in the sense that if you drive for twice the number of hours, then you expect to go twice as far, and if you buy
three times as much stuff, then you expect to pay three times as much sales tax. This section deals with these
kinds 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 k is the constant of proportionality or rate of change.
Here are some examples of everyday proportionality constants:
1. Speed: miles per hour
2. Sales Tax rate: $ of tax per $ of purchase
3. Water flow: gallons per minute
4. Gas: $ per gallon
5. Oatmeal: $ per ounce
6. Wood: $ per foot
7. Monetary conversion: euros per dollar
8. 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?
College Mathematics Notes
Section 2.7
Page 2 of 4
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 meters
has an area of 78.5 square meters, find the area of a sphere with a radius of 5.2 cm. Use the correct
number 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 r and s. If P = 16 when r = 5 and s = -8, find P when r = 2 and s = 10.
College Mathematics Notes
Section 2.7
Page 3 of 4
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 foot
house that is 5.5 miles from the treatment plant is worth $285,000, find the value of a 1.0 acre plot with
a 2,800 square foot house that is 1.75 miles from the plant.
Inverse variation, or inverse proportion is the case where one quantity is proportional to the reciprocal of
another quantity. Examples include: your homework grade is inversely proportional to the number of hours
spent watching TV, or the loudness of a sound you hear is inversely proportional to the square of the distance
from the source.
1
k
y k  y 
x
x
To solve an inverse variation problem:
1) Write down the inverse variation equation: Quantity1 = constant / Quantity2
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:
6. A student who watches 8 hours of TV per week has an average homework score of 94%. If the
homework score is inversely proportional to the number of hours of TV watched, find the average score
of a student who watches 20 hours of TV per week.
College Mathematics Notes
Section 2.7
Page 4 of 4
7. The resistance of a circuit element is inversely proportional to the current flowing through it. If the
current = 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 absolute
temperature (T), and the volume occupied by the gas (V). The relationship between these variables is
that the pressure of a gas varies as the product of the amount of gas and the absolute temperature and
inversely as the volume occupied by the gas. If P is 133,000 Pascals when T is 373 K, n = 120.3 moles,
and V is 2.8 cubic meters, what is P when T is 400. K, V is 1.9 cubic meters, and n = 15.5 moles? Use
the 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. How
fast does the second gear rotate?