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Introduction

This tutorial will guide you through the concepts of simple and compound rules of three, which are essential mathematical tools used to solve proportional problems. Understanding these concepts can significantly enhance your problem-solving skills, especially in exams. We will break down the key principles and provide practical exercises to reinforce your learning.

Step 1: Understanding the Rule of Three

The rule of three is a method for solving problems involving proportional relationships. It can be divided into two types: simple and compound.

Simple Rule of Three

• Definition: Used when two variables are directly proportional.

• Formula: If A is to B as C is to X, then:

  [ X = \frac{B \times C}{A} ]

Example:

• If 2 apples cost $4, how much do 5 apples cost?

  • Set up the equation:
    • 2 (apples) → 4 (dollars)
    • 5 (apples) → X (dollars)
  • Using the formula:

  [ X = \frac{4 \times 5}{2} = 10 ]

Step 2: Applying the Simple Rule of Three

• Identify the known quantities: Write down the values given in the problem.
• Set up the proportion: Place the known quantities in a fraction format.
• Solve for the unknown: Use the formula to find the value of X.

Tips:

• Always ensure that the units are consistent (e.g., apples with apples, dollars with dollars).
• Double-check your calculations to avoid simple arithmetic errors.

Step 3: Understanding the Compound Rule of Three

The compound rule of three is utilized when two or more variables are involved, requiring multiple steps to solve.

Definition

• Compound Rule: Used when there are two sets of variables that influence each other.

Example:

• If 3 workers can complete a job in 6 days, how long will it take 5 workers to complete the same job?

  • Step 1: Calculate the total work in worker-days.
    • Total work = Workers × Days = 3 × 6 = 18 worker-days.
  • Step 2: Determine how many days it will take 5 workers.

  [ \text{Days} = \frac{\text{Total Work}}{\text{Number of Workers}} = \frac{18}{5} = 3.6 \text{ days} ]

Step 4: Applying the Compound Rule of Three

1. Calculate total work: Identify the total work done based on the number of workers and time.
2. Reassign the variables: Change the workers or days in the scenario.
3. Solve the new scenario using the same principles.

Common Pitfalls:

• Misunderstanding the relationship between variables can lead to incorrect setups.
• Forgetting to adjust for changes in scenarios can alter the outcome dramatically.

Conclusion

The rule of three—both simple and compound—is a fundamental concept in mathematics that helps in solving proportional problems effectively. By practicing these steps and applying them to various examples, you'll develop a strong understanding of how to use these rules in real-world situations. For further mastery, consider tackling more complex problems or exploring additional resources on mathematical concepts.