Direct Variation

What is direct variation?

• Two variables are said to vary directly if their ratio is constant (k)
  • This is also called direct proportion
  • If If and  (for positive integer n) vary directly then
  •  for some constant k
  • This can be written as 
  • The graphs of these models always start at the origin

How to solve a problem

Identify which two variables vary directly

It might not be x and y

It could be  and 

• Use the given information to find their constant ratio k
  • Also called constant of proportionality
  • Substitute the given values of x and y into your formula
  • Solve to find k
  • Write the equation which models their relationship
  • 
  • You can then use the equation to solve problems

Worked example

A computer program sorts a list of numbers into ascending order. The time it takes, t milliseconds, varies directly with the square of the number of items, n, in the list. The computer program takes 48 milliseconds to order a list with 8 items.

Find an equation connecting t and n.

b)

Find the time it takes to order a list of 50 numbers.

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Inverse Variation

What is inverse variation?

• Two variables are said to vary inversely if their product is constant (k)
  • This is also called inverse proportion
• If  and  (for positive integer n) vary inversely then
  • It is denoted 
  •  for some constant k
    • This can be written 
      
• The graphs of these models all have a vertical asymptote at the y-axis
  • This means that as x gets closer to 0 the absolute value of y gets further away from 0
  • x can never equal 0

How do I solve inverse variation problems?

• Identify which two variables vary inversely
  • It might not be x and y
  • It could be  and 
• Use the given information to find their constant product k
  • Also called constant of proportionality
  • Substitute the given values of  and  into your formula
  • Solve to find k
• Write the equation which models their relationship
• You can then use the equation to solve problems

Exam Tip

• Reciprocal graphs generally have two parts/curves
  • Only one – usually the positive – may be relevant to the model
  • Think about why x/t/θ can only take positive values - refer to the context of the question

Worked example

The time,  hours, it takes to complete a project varies inversely to the number of people working on it, . If 4 people work on the project it takes 70 hours to complete.

a)

Write an equation connecting t and n.

b)

Given that the project needs to be completed within 18 hours, find the minimum number of people needed to work on it.

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1. Number & Algebra

2. Functions

2.1 Linear Functions & Graphs

2.2 Further Functions & Graphs

2.3 Modelling with Functions

• 2.3.1 Linear & Piecewise Models
• 2.3.2 Quadratic & Cubic Models
• 2.3.3 Exponential Models
• 2.3.4 Direct & Inverse Variation
• 2.3.5 Sinusoidal Models
• 2.3.6 Strategy for Modelling Functions

3. Geometry & Trigonometry

4. Statistics & Probability

5. Calculus

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.