Direct and Inverse Proportion

Identifying Direct and Inverse Proportions

  • A direct proportion occurs when two quantities increase or decrease together at the same rate.
  • An inverse proportion happens when one quantity increases as the other decreases. They change in opposite directions.
  • In a direct proportion, the ratio of the two quantities always stays the same. For inverse proportion, the product of the two quantities is constant.

Solving Problems Involving Direct Proportions

  • Firstly, determine if the problem follows a direct proportion by checking if the ratio remains consistent throughout the quantities.
  • If it’s a direct proportion, you can establish a unitary ratio. This ratio simplifies the proportion to how much one unit would equal.
  • You can then use the unitary ratio to find any multiplicity of that quantity.
  • For instance, if 5 apples cost £2, one apple would cost £0.4 (2/5). If you wanted to find out how much 7 apples cost, multiply 7 by £0.4.

Solving Problems Involving Inverse Proportions

  • Determine if the problem is an inverse proportion by checking whether an increase in one quantity results in a decrease in the other quantity at a consistent rate.
  • In an inverse proportion, the product of the two quantities is always the same.
  • You can therefore use this constant to work out missing or unknown values.
  • For instance, if 6 machines can complete a job in 4 hours (24 machine-hours), then 4 machines would need 6 hours to complete the same job.

Applying Proportional Reasoning

  • Proportional reasoning involves identifying and using mathematical relationships among equivalent ratios.
  • It’s a useful skill for a range of real-world problems, including those in the fields of physics, biology, economics, and engineering.
  • Always remember to check your solutions in context of the problem to validate their reasonableness. For example, a negative time value wouldn’t make sense in a real-life situation.

Using Graphs to Represent Direct and Inverse Proportions

  • A direct proportion can be represented by a straight line graph through the origin (0,0). The steeper the line, the larger the constant of proportionality.
  • An inverse proportion can be represented by a hyperbola on a graph. The steeper the curve, the larger the constant of proportionality.
  • Graphing proportional relationships can help you intuitively understand the behaviour of the quantities involved.