But total = 31/7 × (105/31) = 105 / 7 = 15 → correct. - Richter Guitar

Understanding the Context

At first glance, this may appear obscure, but a closer look uncovers a powerful cancellation principle based on fraction manipulation. Let’s explore how this holds true—and why it’s a great example of simplifying expressions step by step.

Breaking Down the Expression Step by Step

Start with the original expression:
31⁄7 × (105⁄31)

This can be rewritten as:
(31 × 105) ÷ (7 × 31)

Key Insights

Now, notice that numerator contains both 31 and 105, while the denominator contains 31 and 7. Since 31 appears in both numerator and denominator, they cancel out:
= (  × 105  ) ÷ (7 ×  )
= 105 ÷ 7

Performing the division gives:
= 15

Thus, regardless of the initial structure, 31⁄7 × (105⁄31) simplifies directly to 105 ÷ 7, which equals 15.

Why This Identity Matters

This calculation demonstrates a fundamental property of fractions: cancellation via common factors. When multiplying a fraction by another that enthält the same prime factors in the denominator, they effectively cancel, reducing complexity. This principle is essential in algebra, simplifying equations, and solving real-world problems in science, finance, and engineering.

Final Thoughts

Practical Implications of Simplification

Understanding such identities helps streamline mathematical reasoning. For example, in data processing or scaling factors—like adjusting growth rates or unit conversions—canceling common terms early reduces errors and improves efficiency. In this case, reducing 31⁄7 × (105⁄31) avoids carrying large numbers unnecessarily.

Final Takeaway

So, 31⁄7 × (105⁄31) = 15 is not just arbitrary arithmetic—it’s a concise illustration of how careful fraction manipulation leads to clarity and truth. By canceling shared factors, complex-looking formulas simplify elegantly, reinforcing clarity across mathematical applications.

Next time you see a fraction expression like this, remember: look for overlaps, seek cancellation, and let math simplify itself.

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Key Takeaways:

• Mathematic expressions often simplify through cancellation.
  
• Fraction arithmetic relies on identifying and eliminating common factors.
  
• Simplifying 31⁄7 × (105⁄31) reveals 15 efficiently via numerator-denominator cancellation.
  
• Understanding such patterns strengthens problem-solving skills across STEM fields.