MODULE 1

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writer: Daren V. Perez
• Cover Illustrator: Joel J. Estudillo
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• MATHEMATICS Quarter 2: Module 1 Direct and Inverse Variation
• Direct variation


  A mathematical relationship between two variables that can be expressed by an equation in which one variable is equal to a constant times the other
• Inverse variation


  A mathematical relationship between two variables which can be expressed by an equation in which the product of two variables is equal to a constant
• Illustrate situations that involve the following variation:

  1. Direct

  2. Inverse
• Translate into variation statement a relationship between two quantities given by:

  1. Table of values

  2. Mathematical equation

  3. Graph, and vice versa
• y varies directly as x


  Mathematically expressed as y = kx, where k is the constant of variation
• As x increases


  y also increases
• As x decreases


  y decreases
• y varies inversely as x

  Mathematically expressed as y = k/x, where k is the constant of variation
• An increase in x


  Causes a decrease in y
• A decrease in x


  Causes an increase in y
• Direct variation examples


  • Time to hear thunder vs distance from lightning
  • Number of cupcakes vs amount of flour
• Inverse variation examples


  • Number of days to build a house vs number of workers
  • Sounds produced by siren vs distance from ambulance
• A direct variation occurs whenever a situation produces pairs of numbers in which their ratio is constant
• An inverse variation occurs whenever a situation produces pairs of numbers in which their product is constant
• As x increases, y also increases. Similarly, a decrease in x causes a decrease in y
• An increase in x causes a decrease in y and a decrease in x causes an increase in y
• The independent variable is x, the dependent variable is y, and k is the constant of variation
• y = kx, where k is the constant of variation
• y = k/x, where k is the constant of variation
• y varies directly as x


  Mathematically represented as y = kx, where k is the constant of variation
• y varies inversely as x


  Mathematically represented as y = k/x, where k is the constant of variation
• Translating variation statement into:

  1. Table of values

  2. Variation equation

  3. Graph
• The price P varies directly as the number n of oranges is written as P = kn
• An example of inverse variation is y = k/x
• Direct variation is represented by y = kx, inverse variation is represented by y = k/x
• Direct variation


  When two quantities y and x have a constant ratio k = y/x or y = kx, we say that y varies directly as x by a factor k, where k is called the constant of variation
• Inverse variation


  When two quantities y and x have a constant product k = yx or y = k/x, we can say that "y varies inversely as x". The constant product k is also called the constant of variation where x is not equal to zero.
• Solving direct variation problems

  1. Represent the two quantities that relate the problem

  2. Write a mathematical statement and equation between the two quantities

  3. Substitute known values and solve for the constant of variation k

  4. Use the equation to solve for the unknown quantity
• Solving inverse variation problems

  1. Represent the two quantities that relate the problem

  2. Write a mathematical statement and equation between the two quantities

  3. Substitute known values and solve for the constant of variation k

  4. Use the equation to solve for the unknown quantity
• Constant of variation


  A constant that represents the relationship between two variables that vary directly or inversely
• The volume of gas in a container at a constant temperature varies inversely as the pressure
• Data involving the relation between pressure and volume

  • Volume (cm3): 54, 40, 32, 18
  • Pressure (kg): 16/3, 36/5, 9, 16
• Steps to solve the problem

  1. Write the variation equation that represents the relation

  2. Graph the relation

  3. What is the constant of variation?

  4. Find the pressure when the volume of the container is 60 cm3
• If y varies directly as x and y = 26 when x = 1/2, the constant of variation is 52
• If x varies directly as y and x = 54 when y = 9, the value of y when x = 24 is 6