Understanding the Simple Fraction Equality: $ rac{756}{1820} = rac{189}{455} $

When simplifying fractions, one common technique is to reduce them to their lowest terms — and in this article, we explore how $ rac{756}{1820} $ equals $ rac{189}{455} $ through elegant fraction simplification. This process not only reveals the mathematical connection between these two fractions but also demonstrates the power of the greatest common divisor (GCD) in reducing complex ratios.

Understanding the Context

What Are Equivalent Fractions?

Equivalent fractions represent the same value using different numerators and denominators. For example, $ rac{1}{2} $, $ rac{2}{4} $, and $ rac{189}{455} $ are all equivalent expressions of the same quantity. One key way to find equivalent fractions is by simplifying a given fraction to lowest terms.

Simplifying $ rac{756}{1820} $

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Key Insights

To simplify $ rac{756}{1820} $, we begin by finding the greatest common divisor (GCD) of the numerator and denominator.

Step 1: Factor both numbers

  • Prime factorization of 756:
    $ 756 = 2^2 \cdot 3^3 \cdot 7 $

  • Prime factorization of 1820:
    $ 1820 = 2^2 \cdot 5 \cdot 7 \cdot 13 $

Step 2: Identify the common factors

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📰 3(y^3 - 6y^2 + 12y - 8) - 4(y^2 - 4y + 4) + 5y - 10 - 2 = 3y^3 - 18y^2 + 36y - 24 - 4y^2 + 16y - 16 + 5y - 12 = 3y^3 - 22y^2 + 57y - 52. 📰 Thus, $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $. Expanding: 📰 3(x^6 - 6x^4 + 12x^2 - 8) - 22(x^4 - 4x^2 + 4) + 57x^2 - 114 - 52 = 3x^6 - 18x^4 + 36x^2 - 24 - 22x^4 + 88x^2 - 88 + 57x^2 - 166 = 3x^6 - 40x^4 + 181x^2 - 278.

Final Thoughts

The common prime factors are $ 2^2 $ and $ 7 $:

$$
\ ext{GCD}(756, 1820) = 2^2 \cdot 7 = 4 \cdot 7 = 28
$$

Divide numerator and denominator by the GCD

Now divide both the numerator and denominator by 28:

$$
rac{756 \div 28}{1820 \div 28} = rac{27}{65}
$$

Wait — but earlier we claimed $ rac{756}{1820} = rac{189}{455} $. Let’s verify that:

  • $ 756 \div 28 = 27 $
    - $ 1820 \div 28 = 65 $

So $ rac{756}{1820} = rac{27}{65} $

But the claim is $ rac{756}{1820} = rac{189}{455} $ — these appear unequal at first glance.