Anomalous Cancellation on 2-Digit Numbers/Examples/26 over 65

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Example of Anomalous Cancellation on 2-Digit Numbers

The fraction $\dfrac {26} {65}$ exhibits the phenomenon of anomalous cancellation:

$\dfrac {26} {65} = \dfrac 2 5$

as can be seen by deleting the $6$ from both numerator and denominator.


This is part of a longer pattern:

$\dfrac 2 5 = \dfrac {26} {65} = \dfrac {266} {665} = \dfrac {2666} {6665} = \cdots$


Proof

\(\displaystyle \frac {266 \cdots 66} {666 \cdots 65}\) \(=\) \(\displaystyle \paren {2 \times 10^n + \paren {\sum_{i \mathop = 0}^{n - 1} 6 \times 10^i} } \Big / \paren {\paren {\sum_{i \mathop = 1}^n 6 \times 10^i} + 5}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 \times 10^n + 6 \times \paren {\frac {10^n - 1} {10 - 1} } } \Big / \paren {6 \times 10 \times \paren {\frac {10^n - 1} {10 - 1} } + 5}\) Sum of Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2 \times \paren {10 - 1} 10^n + 6 \times \paren {10^n - 1} } {60 \times \paren {10^n - 1} + 5 \paren {10 - 1} }\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {18 \times 10^n + 6 \times 10^n - 6} {60 \times 10^n - 60 + 45}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {24 \times 10^n - 6} {60 \times 10^n - 15}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2 \times \paren {12 \times 10^n - 3} } {5 \times \paren {12 \times 10^n - 3} }\) factoring
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 5\)

$\blacksquare$


Sources