How to Find the Inverse Proportion of a Ratio?

Finding the inverse proportion related to a ratio involves understanding how quantities change in opposite directions. When two quantities are inversely proportional, as one increases, the other decreases proportionally. This relationship can be expressed mathematically and used to solve for unknown values.

Understanding Inverse Proportion

Inverse proportion (also known as inverse variation or reciprocal proportion) means that two variables are related in such a way that their product is a constant. If we call the variables x and y, and the constant k, then the relationship is:

x y = k

This means if x doubles, y halves, and vice-versa.

Steps to Find the Inverse Proportion

1. Identify the Variables and their Initial Values: Determine the two quantities that are inversely proportional and note their initial values. Let's say x₁ and y₁ are the initial values of x and y, respectively.

2. Calculate the Constant of Proportionality (k): Use the initial values to find the constant k using the formula:

   k = x₁ y₁

3. Determine the New Value of One Variable: If you are given a new value for one variable (let's say x₂), you can find the corresponding value of the other variable (y₂).

4. Calculate the Inverse Proportion Value: Use the constant k and the new value to solve for the unknown variable:

   y₂ = k / x₂

   or

   x₂ = k / y₂

Example

Let's say that the time it takes to complete a task is inversely proportional to the number of workers. If 4 workers can complete a task in 6 hours, how long will it take 8 workers to complete the same task?

1. Variables:

   • x = Number of workers
   • y = Time to complete the task

   Initial values: x₁ = 4, y₁ = 6

2. Calculate k:

   • k = x₁ y₁ = 4 * 6 = 24

3. New Value:

   • x₂ = 8

4. Calculate the Inverse Proportion Value:

   • y₂ = k / x₂ = 24 / 8 = 3

Therefore, it will take 8 workers 3 hours to complete the task.

In Summary

To find the inverse proportion, calculate the constant k using initial values, then use that constant and a new value of one variable to find the corresponding value of the other variable. This applies the fundamental relationship of inverse proportionality: as one value increases, the other decreases proportionally, maintaining a constant product.