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Proportional Difference

Problem

It can be seen that $\dfrac{12}{20} = \dfrac{3}{5} = \dfrac{12 - 3}{20 - 5} = \dfrac{9}{15}$.

If $\dfrac{a}{b} = \dfrac{c}{d}$ and $a \ne c$ then prove that $\dfrac{a - c}{b - d} = \dfrac{a}{b}$.


Solution

If $\dfrac{a}{b} = \dfrac{c}{d}$ then let $c = ka$ and $d = kb$.

$\therefore \dfrac{a - c}{b - d} = \dfrac{a - ka}{b - kb} = \dfrac{a(1 - k)}{b(1 - k)} = \dfrac{a}{b}$   Q. E. D.

Problem ID: 314 (18 Mar 2007)     Difficulty: 2 Star

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