TOPICS

Keywords

• Rule of three
• Inverse proportion
• Inversely proportional quantities
• Ratio
• Dependency relation
• Cross multiplication

Key Questions

• How to identify an inverse proportion relationship between two quantities?
• What are the steps to solve problems involving indirect rule of three?
• How does the variation of one quantity affect the other in an inversely proportional relationship?

Crucial Topics

• Understanding of indirect proportion/increase-decrease
• Definition of inversely proportional quantities
• Strategy of value inversion to solve indirect rule of three
• Application in everyday problems and in various areas of knowledge

Formulas

The relationship between two inversely proportional quantities is given by the multiplication of the means and extremes (cross product), resulting in a constant.

Let x and y be inversely proportional quantities, and a1, a2, b1, b2 the corresponding values, we have:

a1 * b1 = a2 * b2

• Example of applying the formula in indirect rule of three:

If 6 workers take 8 hours to complete a task, how many hours would it take for 4 workers?

6 workers — 8 hours
4 workers — x hours

We apply the inverse proportion relationship:

6 * 8 = 4 * x x = (6 * 8) / 4 x = 12 hours

NOTES

• Key Terms

  • Rule of three: Mathematical method to determine an unknown fourth value when three values are known in two proportional quantities.
  • Inverse proportion: Occurs when an increase in one quantity results in the decrease of the other in the same ratio, and vice versa.
  • Inversely proportional quantities: Two quantities are inversely proportional if the product of their corresponding values is constant.
  • Ratio: Relationship between two numbers indicating how many times one contains the other.
  • Dependency relation: Indicates how one quantity varies depending on the other.
  • Cross multiplication: Tool used to solve proportions, equalizing products of opposite terms (means and extremes).

• Main Ideas, Information, and Concepts

  • Understanding indirect proportion is vital to solve indirect rule of three problems.
  • Identifying when quantities are inversely proportional is the first step to correctly apply the indirect rule of three.
  • Using the strategy of value inversion allows solving indirect rule of three problems, transforming it into a direct proportion.

• Topic Contents

  • To solve an indirect rule of three problem, one must first establish the inverse proportion between the quantities.
  • After identification, it is necessary to invert one of the value columns so that the direct proportion is applied.
  • The solution involves cross multiplication and solving an equation to find the unknown value.

• Examples and Cases

  • Example 1: If a car consumes 8 liters of fuel to travel 100 km, how many liters will it consume to travel 150 km considering an inverse proportion of speed and consumption?
    • First, identify that the higher the speed, the less the car consumes in relation to the distance.
    • Then, invert the liters column to transform it into a direct proportion.
    • After applying cross multiplication, solve to find the unknown value.
  • Example 2: If a tap fills a tank in 3 hours, how long would it take for two identical taps to fill the same tank?
    • Establish that the number of taps and time are inversely proportional quantities.
    • Invert the taps column to set up the direct proportion.
    • Apply cross multiplication and find the new time required to fill the tank.

LESSON SUMMARY

• Indirect rule of three: A technique to find an unknown value when two quantities are inversely proportional.

• Identification of inversely proportional quantities: Recognize when an increase in one quantity causes a decrease in another and apply the rule of three by inverting one of the value columns.

• Cross multiplication: Applying cross multiplication to find the unknown value, respecting the inverse proportionality.

Conclusions

• The indirect rule of three is used when two quantities have an inverse dependency relationship.
• To solve this type of problem, it is crucial to invert the values of one of the quantities to transpose the situation into a direct proportion.
• The correct application of cross multiplication is essential to reach the desired result.
• Indirect rule of three problems are present in various everyday situations and interdisciplinary contexts, requiring analytical skills and application of mathematical concepts.