MODULE 2

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writer: Randolf Byron S. Viray
• Cover Illustrator: Joel J. Estudillo
• MATHEMATICS Quarter 2: Module 2 Joint and Combined Variations
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• Joint Variation


  A relationship among three variables that can be written in the form y = kxz, where k is a constant of variation or constant of proportionality
• Combined Variation


  A variation that involves combinations of direct, inverse and joint variation. It can be expressed in the form y = kx/z, where k is the constant of variation or the constant of proportionality
• Steps in Solving Joint and Combined Variation

  1. Write the correct mathematical equation

  2. Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality

  3. Rewrite the equation from step 1 by substituting in the value of k found in step 2

  4. Use the equation found in step 3 and the remaining information given in the problem to answer the question asked
• Illustrative Examples for Joint Variation

  • y = 24 when x = 2 and z = 3
  • y = 12 when x = 6 and z = 4
  • z = -75 when x = 3 and y = -5
• Illustrative Examples for Combined Variation


  • a = 12 when b = 2 and c = 3
  • y = 14 when x = 7 and z = 8
• The statement "y varies directly as x and z" denotes joint variation
• The statement "y varies directly as x and inversely as z" denotes combined variation
• Varies directly


  One quantity increases in proportion to the increase in another quantity
• Varies inversely


  One quantity decreases in proportion to the increase in another quantity
• Varies jointly

  Two or more quantities vary directly with each other
• Varies combined


  One quantity varies directly with one variable and inversely with another variable
• Steps to solve joint and combined variation problems

  1. Write the correct variation equation

  2. Use the given information to find the constant of variation (k)

  3. Substitute the known values into the variation equation to find the missing value
• Leonhard Euler was a Swiss mathematician and physicist who invented the calculus of variation
• Discipline

  Good Taste
• The acceleration due to gravity is represented by the symbol g
• The number of Fruits F in a row

  Varies inversely as the space s between them
• The area A of a triangle is proportional to its height h
• c

  Proportional to w and L
• c

  Proportional to wL
• Translate variation statement into variation equation


  The appropriate length s of a rectangular beam varies jointly as its width w and its depth d
• s

  Proportional to kwdC. s = kwdD. s = wd
• Translate variation statement into variation equation


  The drag force F on a boat varies jointly with the wet surface area A of the boat and the square of the speed s of the boat
• Variation Equation


  • A = kwL2. F varies directly as r and the square of h and inversely as q3. S = ku/h^24. A = kg/t5. A is jointly proportional to h, cube of r, and square of b
• Crossword Puzzle
• Organization


  Information appears to be disorganized<|>Information is organized, but not entirely sequential or with clarity<|>Information is organized and sections are identified with a good amount of clarity and sequencing<|>Information is clear and very well organized with each section in clear sequential order
• Explanation


  Student poorly explained his thinking<|>Student somehow explained his thinking using one explanation with very minimal evidence of thinking<|>Student explained his thinking using one explanation and provided evidence of thinking<|>Student explained and supported his thinking using more than one explanation, and provided evidence of thinking
• Joint Variation


  The amount of Simple Interest (I) earned in an account varies jointly as the amount of principal (P) invested and the amount of time(T) the money is invested
• Php 5,000 in principal earns Php 750 in 6 years
• Solving Joint Variation problems


  1. Write the equation

  2. Find the constant of variation

  3. Solve for the unknown
• Combined Variation


  The number of minutes (Y) needed to solve an exercise set of variation problems varies directly as the number of problems (X) and inversely as the number of people (Z) working on the solutions
• It takes 4 people 36 minutes to solve 18 problems
• Solving Combined Variation problems


  1. Write the equation

  2. Find the constant of variation

  3. Solve for the unknown
• Volume of gas


  Varies directly as the temperature and inversely as the pressure
• Volume is 250 cm3 when the temperature is 400°K and the pressure is 20 lb/cm2
• Direct Variation


  A quantity depends directly on one or more other quantities