Question
If $a : b : : c : d,$ then prove that
$2a+7b : 2a-7b = 2c+7d : 2c-7d$
$2a+7b : 2a-7b = 2c+7d : 2c-7d$
✓
Answer
$2 a +7 b : 2 a -7 b =2 c +7 d : 2 c -7 d $
$\frac{ a }{ b }=\frac{ c }{ d } $
$\text { Multiplying both sides by } \frac{2}{7} $
$\Rightarrow \frac{ a }{ b } \times \frac{2}{7}=\frac{ c }{ d } \times \frac{2}{7} $
$\Rightarrow \frac{2 a }{7 b }=\frac{2 c }{7 d }$
Applying componendo and dividendo,
$\frac{2 a +7 b }{2 a -7 b }=\frac{2 c +7 d }{2 c -7 d }$
Hence, $2 a+7 b: 2 a-7 b=2 c+7 d: 2 c-7 d$
$\frac{ a }{ b }=\frac{ c }{ d } $
$\text { Multiplying both sides by } \frac{2}{7} $
$\Rightarrow \frac{ a }{ b } \times \frac{2}{7}=\frac{ c }{ d } \times \frac{2}{7} $
$\Rightarrow \frac{2 a }{7 b }=\frac{2 c }{7 d }$
Applying componendo and dividendo,
$\frac{2 a +7 b }{2 a -7 b }=\frac{2 c +7 d }{2 c -7 d }$
Hence, $2 a+7 b: 2 a-7 b=2 c+7 d: 2 c-7 d$
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