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Componendo and Dividendo

If a-2b-3c+4d...

Question

If $\frac{a - 2 b - 3 c + 4 d}{a + 2 b - 3 c - 4 d}$ = $\frac{a - 2 b + 3 c - 4 d}{a + 2 b + 3 c + 4 d}$

Show that : 2ad = 3ab.

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Solution

$fraction numerator a minus 2 b minus 3 c plus 4 d over denominator a plus 2 b minus 3 c minus 4 d end fraction equals fraction numerator a minus 2 b plus 3 c minus 4 d over denominator a plus 2 b plus 3 c plus 4 d end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma fraction numerator left parenthesis a minus 2 b minus 3 c plus 4 d right parenthesis plus left parenthesis a plus 2 b minus 3 c minus 4 d right parenthesis over denominator left parenthesis a minus 2 b minus 3 c plus 4 d right parenthesis minus left parenthesis a plus 2 b minus 3 c minus 4 d right parenthesis end fraction equals fraction numerator left parenthesis a minus 2 b plus 3 c minus 4 d right parenthesis plus left parenthesis a plus 2 b plus 3 c plus 4 d right parenthesis over denominator left parenthesis a minus 2 b plus 3 c minus 4 d right parenthesis minus left parenthesis a plus 2 b plus 3 c plus 4 d right parenthesis end fraction fraction numerator 2 left parenthesis a minus 3 c right parenthesis over denominator 2 left parenthesis negative 2 b plus 4 d right parenthesis end fraction equals fraction numerator 2 left parenthesis a plus 3 c right parenthesis over denominator 2 left parenthesis negative 2 b minus 4 d right parenthesis end fraction fraction numerator a minus 3 c over denominator a plus 3 c end fraction equals fraction numerator negative 2 b plus 4 d over denominator negative 2 b minus 4 d end fraction A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o fraction numerator a minus 3 c plus a plus 3 c over denominator a minus 3 c minus a minus 3 c end fraction equals fraction numerator negative 2 b plus 4 d minus 2 b minus 4 d over denominator negative 2 b plus 4 d plus 2 b plus 4 d end fraction fraction numerator 2 a over denominator negative 6 c end fraction equals fraction numerator negative 4 b over denominator 8 d end fraction fraction numerator a over denominator negative 3 c end fraction equals fraction numerator negative b over denominator 2 d end fraction 2 a d equals 3 b c$

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\frac{a - 2 b - 3 c + 4 d}{a + 2 b - 3 c - 4 d} = \frac{a - 2 b + 3 c - 4 d}{a + 2 b + 3 c + 4 d}

2 a d = 3 b c

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