What is a proportion in math

Here you will learn what a proportion is, including what direct and inverse proportions are and how to solve problems with them.

Students will first learn about proportions in math as part of ratios and proportions in katex is not definedth grade.

What is a proportion in math?

A proportion in math expresses the equivalence of two ratios.

A direct proportion in math is represented by the equation katex is not defined

This means that katex is not defined is proportional to katex is not defined is a constant value that links the two variables, called the constant of proportionality.

When katex is not defined increases as katex is not defined increases, the proportion is called a direction variation or direct proportion.

For example,

If katex is not defined then katex is not defined would be double the value of katex is not defined for every value of katex is not defined

​​Direct proportions are seen in many real life situations such as exchange rates, converting between units, and the price of something per dollar.

Step-by-step guide: Direction variation equation

Step-by-step guide: Directly proportional

Step-by-step guide: Direction variation

Inverse proportionality describes a relationship between katex is not defined and katex is not defined where an increase in katex is not defined results in a decrease in katex is not defined

For example, the time taken to paint a house is inversely proportional to the number of workers. As the number of workers increases, the painting time decreases.

An inverse proportion in math is represented by the equation katex is not defined where the numerator katex is not defined is the constant of proportionality and the denominator katex is not defined is the independent variable.

Step-by-step guide: Direct and inverse variation

Step-by-step guide: Inversely proportional

Step-by-step guide: Inverse proportion formula

Proportional relationships can be graphed. When graphing the equation katex is not defined where katex is not defined is a constant, the graph is a straight line passing through the origin.

However, when we consider the reciprocal function katex is not defined the relationship between katex is not defined and katex is not defined becomes inverse.

As katex is not defined increases, the value of katex is not defined decreases. This results in a curved graph that does not pass through the origin nor intersects either axis.

Step-by-step guide: Directly proportional graph

What is a proportion in math?

Common Core State Standards

How does this relate to katex is not definedth grade math?

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2)
  Recognize and represent proportional relationships between quantities.
  
  a. Decide whether two quantities are in a proportional relationship, for example, by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
  
  b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
  
  c. Represent proportional relationships by equations. For example, if the total cost katex is not defined is proportional to the number of items katex is not defined purchased at a constant price katex is not defined the relationship between the total cost and the number of items can be expressed as katex is not defined
  
  d. Explain what a point katex is not defined on the graph of a proportional relationship means in terms of the situation, with special attention to the points katex is not defined and katex is not defined where katex is not defined is the unit rate.

• Grade 7 – Ratios and Proportional Relationships (7.RP.A.2b)
  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• Grade 8 – Expressions and Equations (8.EE.B.5)
  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

• 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 solve problems with a proportion in math

There are a lot of ways to solve problems with a proportion in math. For more specific step-by-step guides, check out the pages linked in the “What is a proportion in math?” section above or read through the examples below.

What is a proportion in math examples

Example 1: write a proportional equation

If katex is not defined and katex is not defined vary directly, write the direct variation equation when katex is not defined and katex is not defined

1. Calculate the constant of variation, katex is not defined

To calculate katex is not defined use the direct variation equation, katex is not defined

katex is not defined can also be written as katex is not defined

Substitute in katex is not defined for katex is not defined and katex is not defined for katex is not defined to calculate katex is not defined

The constant of variation katex is not defined is katex is not defined

2Write the equation in the form of katex is not defined

The direct variation equation can be written by substituting katex is not defined in for katex is not defined

Example 2: unknown variable in a proportion table

Given that katex is not defined is directly proportional to katex is not defined calculate the value for katex is not defined when katex is not defined

Example 3: find the constant of variation from a graph

Use the graph to find the constant of variation.

Example 4: complete the table y ∝ 1/x

Let katex is not defined Calculate the value for katex is not defined when katex is not defined

Example 5: determine the inverse proportion formula when y ∝ 1/x

Let katex is not defined be inversely proportional to katex is not defined When katex is not defined By calculating the constant of proportionality, determine a formula for katex is not defined in terms of katex is not defined

Example 6: using a directly proportional graph

The graph shows the relationship between the US Dollar and Mexican Pesos. Use the graph to estimate the number of US Dollars there are for katex is not defined Mexican Pesos.

Locate the value on the correct axis.

Show step

You have been given the value katex is not defined Mexican Pesos, so you need to locate this value on the vertical axis.

Use straight lines to find the corresponding value from the other axis.

Show step

Draw a straight line across the graph and then another straight line down to the other axis.

Teaching tips for what is a proportion in math

• Before teaching and solving proportions, review topics such as equivalent fractions, how to simplify fractions or a given ratio.

• Start with whole numbers before moving on to integers and fractional numbers in proportion problems.

• Choose worksheets that have different types of proportions shown in different ways, including word problems, real-life conversion, and continued proportions.

Easy mistakes to make

• Incorrectly using cross multiplication
  The cross multiply strategy can be useful in certain instances if students understand why it works. It is important for students to have enough meaningful experiences with proportions to develop an understanding of all properties of proportions.
  
  Conversely, when students do not understand why the cross products strategy works, they can make mistakes such as multiplying only one side of the equation or writing the proportion formula down incorrectly.

• Thinking proportions only increase
  Since students are typically first introduced to direct proportions, this is understandable. As students advance in their understanding of inverse proportions, students are exposed to the decrease and can update this line of thinking.

• Substituting the katex is not defined-coordinate and katex is not defined-coordinate incorrectly
  When you are given a point and need to find the equation of direct variation, substitute the katex is not defined-coordinate in for the katex is not defined value and the katex is not defined-coordinate for the katex is not defined-value.
  
  For example, katex is not defined
  
  

The next lessons are

• Converting fractions decimals and percentages
• Percent
• Compound measure

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