Direct and Inverse Proportions

Understanding Direct and Inverse Proportions

  • Direct proportion is when two variables increase or decrease at the same rate.
  • The constant of proportionality, often denoted by the letter ‘k’, is the factor by which the two values change in direct proportion.
  • For example, if you’re driving at a constant speed, the distance you travel (d) is directly proportional to the time you travel (t). Hence d = kt, where k is your constant speed.
  • Inverse proportion describes a relationship where an increase in one variable causes a decrease in another, and vice versa.
  • This can also be articulated as ‘varies inversely’ or ‘inversely proportional’. It is usually written as y = k/x, where ‘k’ is the constant of proportionality.

Direct Proportion Calculations

  • To handle calculations involving direct proportions, set up an equation using the information given.
  • Ensure to replace the ‘is proportional to’ statement with ‘= k *’ within your equation.
  • Next, use the given values to find ‘k’ first. Once ‘k’ has been discovered, it can be used to find other unknowns.

Inverse Proportion Calculations

  • In calculations involving inverse proportions, start by writing your equation as y = k/x.
  • To find ‘k’, multiply the two values given. This makes ‘k’ a constant for any given question.
  • Once ‘k’ is found, it can be used to figure out other unknown values.

Practice Problems

  • Problem: If y is directly proportional to x, and y = 12 when x = 3, what is the value of y when x = 5?
    • Solution: Find k first, k = y/x = 12/3 = 4. Hence when x = 5, y = 5 * 4 = 20.
  • Problem: If p varies inversely as q, and p = 2 when q = 4, what is the value of p when q = 8?
    • Solution: Here, k = pq = 24 = 8. So when q = 8, p = k/q = 8/8 = 1.

Final Notes

  • Understand that ‘direct proportion’ means an increase in one variable corresponds to an increase in another, while ‘inverse proportion’ indicates that an increase in one leads to a decrease in another.
  • Always find your constant of proportionality ‘k’ early in your calculations.
  • Highlight the difference between the two equations: y = kx for direct proportion and y = k/x for inverse proportion.
  • It’s important to master how to use ‘k’ to find other unknown values in both types of proportions.