Anomalous Cancellation/Examples/3544 over 7531

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Theorem

The fraction $\dfrac {3544} {7531}$ exhibits the phenomenon of anomalous cancellation:

$\dfrac {3544} {7531} = \dfrac {344} {731}$

as can be seen by deleting the $5$ from both numerator and denominator.


This is part of a longer pattern:

$\dfrac {344} {731} = \dfrac {3544} {7531} = \dfrac {35544} {75531} = \cdots$


Proof

\(\displaystyle \frac {355 \cdots 544} {755 \cdots 531}\) \(=\) \(\displaystyle \paren {3 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {\sum_{i \mathop = 2}^{n - 1} 5 \times 10^i} + 31}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {3 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \sum_{i \mathop = 0}^{n - 3} \times 10^i} + 31}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {3 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 44} \Big / \paren {7 \times 10^n + \paren {500 \times \frac {10^{n - 2} - 1} {10 - 1} } + 31}\) Sum of Geometric Progression
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 44 \times \paren {10 - 1} } {7 \times 10^n \times \paren {10 - 1} + 500 \times \paren {10^{n - 2} - 1} + 31 \times \paren {10 - 1} }\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {27 \times 10^n + 500 \times 10^{n - 2} - 104} {63 \times 10^n + 500 \times 10^{n - 2} - 221}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {2700 \times 10^{n - 2} + 500 \times 10^{n - 2} - 104} {6300 \times 10^{n - 2} + 500 \times 10^{n - 2} - 221}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3200 \times 10^{n - 2} - 104} {6800 \times 10^{n - 2} - 221}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {8 \times \paren {400 \times 10^{n - 2} - 13} } {17 \times \paren {400 \times 10^{n - 2} - 13} }\) factoring
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 8 {17}\) factoring
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {8 \times 43} {17 \times 43}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {344} {731}\)

$\blacksquare$


Sources