Inverse Variation

• Inverse variation


  A situation that produces pairs of numbers whose product is constant
• Increase in x


  Decrease in y
• Inverse variation statement


  y varies inversely as x or y is equal to k over x
• Representing inverse variation relationship

  1. Translate to mathematical equation y = k/x

  2. Find constant k by multiplying x and y values

  3. Substitute values to solve for y
• Inverse variation statements


  • The number of pizza slices p varies inversely as the number of persons n sharing a whole pizza
  • The length l of a rectangle varies inversely as its width w
  • The mass m of an object varies inversely as the acceleration due to gravity g
• Finding equation and constant for inverse variation

  1. Given: y varies inversely as x, and y = 6 when x = 18

  2. Step 1: Write equation y = k/x

  3. Step 2: Find constant k by multiplying x and y: k = x*y = 18*6 = 108

  4. Step 3: Write final equation y = 108/x
• Finding equation and constant for inverse variation


  1. Given: a varies inversely with b, a = 1/2 when b = 4

  2. Step 1: Write equation a = k/b

  3. Step 2: Find constant k by multiplying a and b: k = a*b = 1/2 * 4 = 2

  4. Step 3: Write final equation a = 2/b
• Finding equation and constant for inverse variation from table of values

  1. Step 1: Identify that y varies inversely as x

  2. Step 2: Write equation y = k/x

  3. Step 3: Find constant k by multiplying x and y for any row: k = x*y

  4. Step 4: Write final equation y = k/x
• Solving inverse variation problem

  1. Given: y varies inversely as x, y = 10 when x = 2

  2. Step 1: Write equation y = k/x

  3. Step 2: Find constant k by multiplying x and y: k = x*y = 2*10 = 20

  4. Step 3: Substitute values to solve for y when x = 10: y = 20/10 = 2
• Inverse variation occurs whenever a situation produces pairs of numbers whose product is constant
• For quantities x and y, an increase in x causes a decrease in y or vice versa
• We can say that y varies inversely as x or y is equal to k over x
• The statement "y varies inversely to x" translates to y is equal to k over x, where k is the constant of variation
• Table of values for distance, speed and time

  • Distance = 40 km, Time = 50 min, Speed = 48 km/h
  • Distance = 40 km, Time = 40 min, Speed = 60 km/h
  • Distance = 40 km, Time = 30 min, Speed = 80 km/h
  • Distance = 40 km, Time = 20 min, Speed = 120 km/h
• Increase in speed


  Decrease in time
• Mathematical statement representing relationship between speed and time

  Speed s is inversely proportional to time t, or s = k/t, where k is the constant
• The constant k in the equation s = k/t is 40, as shown by the examples in the table
• Equation for statement "the number of pizza slices p varies inversely as the number of persons n sharing a whole pizza"

  p = k/n
• Equation for statement "the length l of a rectangular varies inversely at its width w"

  l = k/w
• Equation for statement "the mass m of an object varies inversely as the acceleration due to gravity g"

  m = k/g
• The number of hours t required to finish a job varies inversely as the number of persons n working on the job
• If 16 persons require 18 hours to finish a job, how long would it take 64 persons to finish the job?
• Solving inverse variation problem for job completion time

  1. Step 1: Write equation t = k/n, where k = t*n = 16*18 = 288

  2. Step 2: Substitute values to solve for t when n = 64: t = 288/64 = 4.5 hours