Question

If a + 2b varies as a-2b, prove that a varies as b

Answer 1

If a + 2b varies as a-2b, prove that a varies as b

• a+2b:a−2b::k:1a+2b:a−2b::k:1 implies a+2b=k(a−2b)a+2b=k(a−2b). Therefore a(k−1)=2b(k+1)a(k−1)=2b(k+1) so a:b::k−1:2k+1a:b::k−1:2k+1.

Answer 2

SOLUTION

GIVEN

$\displaystyle\sf{ \:(a + 2b) \propto \: (a - 2b) }$

TO PROVE

$\displaystyle\sf{ \:a \propto \: b }$

EVALUATION

Here it is given that

$\displaystyle\sf{ \:(a + 2b) \propto \: (a - 2b) }$

$\displaystyle\sf{ \implies \frac{(a + 2b)}{(a - 2b)} = constant \: , \: say \: \: k }$

$\displaystyle\sf{ \implies \frac{(a + 2b)}{(a - 2b)} = \frac{k}{1} }$

Using Componendo Dividendo Rule we get

$\displaystyle\sf{ \implies \frac{(a + 2b + a - 2b)}{(a + 2b - a + 2b)} = \frac{k + 1}{k - 1} }$

$\displaystyle\sf{ \implies \frac{2a}{4b} = m \: \: (say) }$

$\displaystyle\sf{ \implies \frac{a}{2b} = m \: }$

$\displaystyle\sf{ \implies \frac{a}{b} = 2m \: }$

$\displaystyle\sf{ \implies \frac{a}{b} = constant \: }$

$\displaystyle\sf{ \implies a \propto \: b }$

Hence proved

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