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Ratio and Proportion

If a,b,c,d ...

Question

If $a, b, c, d$ are in proportion,prove that:
(i) $(5 a + 7 b) (2 c - 3 d) = (5 c + 7 d) (2 a - 3 b)$

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Solution

$a, b, c, d$ are in proportion
$\Rightarrow a d = b c$

Now
LHS:
$(5 a + 7 b) (2 c - 3 d)$
$= 10 a c - 15 a d + 14 b c - 21 b d$
$= 10 a c - 21 b d - a d$

RHS
$(5 c + 7 d) (2 a - 3 b)$
$= 10 a c - 15 b c + 14 a d - 21 b d$
$= 10 a c - 21 b d - a d$

$\Rightarrow L H S = R H S$

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