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Question Number 28554 by naka3546 last updated on 27/Jan/18

\frac{(a - b)}{(c - d)} = 3

\frac{(a - c)}{(b - d)} = 4

\frac{(a - d)}{(b - c)} = ?

Answered by ajfour last updated on 27/Jan/18

a - b = 3 (c - d) \dots (i)

a - c = 4 (b - d) \dots (i i)

(i i) - (i) :

\Rightarrow b - c = 3 (b - c) + b - d

o r 2 (b - c) = d - b \dots (I)

(i) - 3 (i i) :

- 2 a - b + 3 c = 3 c - 3 d - 12 b + 12 d

\Rightarrow - 2 (a - d) = - 11 (b - d) \dots (I I)

(I I) \div (I) :

\frac{a - d}{b - c} = - 11 .

Answered by mrW2 last updated on 27/Jan/18

a - b = 3 c - 3 d \dots (i)

a - c = 4 b - 4 d

a - 4 b = c - 4 d \dots (i i)

(i) - (i i)

3 b = 2 c + d

\Rightarrow b = \frac{2 c + d}{3}

\Rightarrow a = b + 3 c - 3 d = \frac{2 c + d + 9 c - 9 d}{3} = \frac{11 c - 8 d}{3}

a - d = \frac{11 c - 8 d - 3 d}{3} = \frac{11 (c - d)}{3}

b - c = \frac{2 c + d - 3 c}{3} = - \frac{(c - d)}{3}

\Rightarrow \frac{a - d}{b - c} = \frac{\frac{11 (c - d)}{3}}{- \frac{(c - d)}{3}} = - 11

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