Direct and inverse variation

Here you will learn about direct and inverse variation, including the concepts of direct proportion and inverse proportion. You will look at solving some real-life word problems involving these different proportional relationships.

Students will first learn about direct and inverse variation as part of ratios and proportional relationships in 7th grade.

What are direct and inverse variation?

Direct and inverse variation are two different types of proportional relationships that describe how quantities are related to each other.

• Direct variation, or direct proportion, is a relationship between two quantities in which, as one quantity increases, the other quantity also increases.

For example,

The cost of a banana is 70 cents. As the number of bananas increases, so does the cost; 3 bananas would cost 3 times the cost of one banana (\$2.10).

If y is directly proportional to x (y\propto{x}), then y=kx where k is the constant of proportionality.

Step-by-step guide: Directly proportional

• Inverse variation, or inverse proportion, is a relationship between two quantities in which one quantity increases while the other decreases, and vice versa. This is also called indirect variation.

For example, it takes 1 worker 9 hours to dig a hole. As the number of workers increases, the number of hours it takes to dig the same hole decreases. 3 workers would take a third of the time ( 3 hours).

To calculate inverse variation problems, you need to appreciate that multiplication and division are inverse operations of each other.

If y is inversely proportional to x (y\propto\cfrac{1}{x}), then y=\cfrac{k}{x} where k is the constant of proportionality.

Step-by-step guide: Inversely proportional

What are direct and inverse variation?

Common Core State Standards

How does this relate to 7th grade math and high school math?

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2)
  Recognize and represent proportional relationships between quantities.

• High School – Functions – Interpreting Functions (HS.F.IF.B.4)
  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
  
  Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

How to use direct and inverse variation

In order to answer word problems involving direct and inverse variation:

1. Determine the type of proportionality relationship between the two quantities.
2. Calculate the constant of proportionality, \textbf{k} .
3. Calculate the unknown value.

Direct and inverse variation examples

Example 1: direct proportion

A t-shirt costs \$4. How much do 5 t-shirts cost?

1. Determine the type of proportionality relationship between the two quantities.

As the number of t-shirts increases, so does the cost. This is an example of direct variation, or direct proportion.

2Calculate the constant of proportionality, \textbf{k} .

For direct proportion, the constant of proportionality k is the cost of one t-shirt.

As this is a given value (a t-shirt costs \$4 ) you can say k=4 and so y=4x where the y value would represent the total cost of x number of t-shirts.

3Calculate the unknown value.

Substituting x=5 into y=4x, you have y=4\times{5}=20.

4Write the solution.

The cost of 5 t-shirts is \$20.

Example 2: direct proportion

7 bags of candy weigh 350 grams. How much do 10 bags of candy weigh?

Example 3: direct proportion

8 laps of a race track has a total of 12 \, km. What would the distance be for 20 laps of the race track?

Example 4: inverse proportion

A worker takes 10 days to fit a bathroom. How long would it take 2 workers to fit a bathroom?

Example 5: inverse proportion

An oil tank takes 25 hours to be filled by 3 hose pipes. How long does it take 5 hose pipes to fill the same oil tank?

Example 6: inverse proportion

10 computers can do a task in 15 minutes. How long does it take 3 computers to do the same task?

Teaching tips for direct and inverse variation

• Start by explaining that the constant of variation is a fixed number that defines the relationship between two variables in both direct and inverse variation.

• Draw graphs to show how quantities relate in direct variation (a straight line passing through the origin) and in inverse variation (a hyperbola). Tables can also help students see patterns in values.

• Teach students to apply inequalities to explore limits in direct and inverse variation.
  
  For example, in direct variation y=3x, setting x\leq{10} shows that y grows proportionally up to a maximum.
  
  In inverse variation y=\cfrac{12}{x}, using x>0 highlights that as x increases, y decreases but never reaches zero.

• Provide worksheets with a variety of practice questions on direct and inverse variation. Start with straightforward problems to identify constants of variation, then include word problems that require students to set up and solve equations, ensuring they understand how to manipulate both sides of the equation.

• To integrate more advanced concepts, challenge students to work with rational expressions and polynomial equations involving variables x and y, where y increases with changes in x.
  
  Incorporate problems that require them to use square roots and exponents, helping them see the connections between these concepts and variation principles.

Easy mistakes to make

• Modeling assumption
  Whenever you solve word problems for proportion, you assume everything has the same value. If the question involves the costs of pencils, you assume each pencil costs the same. If the question involves the number of people working, you assume all the workers work at the same rate.

• Misunderstanding inverse proportion
  Direct proportion is described as “as one value increases, so does the other.” Inverse proportion is therefore considered to be the opposite where “as one value decreases, so does the other”.
  
  This is not true. An inversely proportional or inverse relationship shows that when one value increases, the other decreases. As a graph, this would look like a reciprocal graph.

• Mistaking an inverse variation problem for a direct variation problem
  For example, if 3 people take 12 hours to build a wall, 6 people take 24 hours to build the same size wall. This is not true as you assume everyone works at the same rate and so the wall should be built in less time if more people are building it.
  
  As the number of people increases, the time taken to build the wall decreases and so if you have 6 builders (double the original amount), the time it takes to build the wall should be 6 hours (half of the original amount). The type of proportion must be determined for every proportionality question.

• Writing money incorrectly
  Money is used in some proportional word problems. If an answer is 4.1 you may be tempted to write it as \$4.1, but the correct way of writing it would be \$4.10.

• Writing time incorrectly
  Time is used in some proportional word problems. If an answer is 7.25 you may be tempted to write it as 7 hours 25 minutes, but it would be 7 hours 15 minutes. (Remember there are 60 minutes in an hour).

Related proportion lessons

• Direct variation equation
• Directly proportional
• Direct variation
• Inversely proportional
• Inverse proportion formula
• Directly proportional graph

The next lessons are

• Converting fractions decimals and percents
• Percent
• Compound measures

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