Question

If $\frac{a+3d}{a+9d}=\frac{a+d}{a+5d}=k$, then $k$ is equal to $(a,d>0)$

• A. $\frac{1}{2}$
• B. $2$
• C. $6$
• D. $0$

Answer 1

Answer: (A)

$\frac{1}{2}$

$\frac{a+3d}{a+9d}=\frac{a+d}{a+5d}$

Applying Componendo-Dividendo

$\Rightarrow \frac{a+3d+a+9d}{a+3d-a-9d}=\frac{a+d+a+5d}{a+d-a-5d}$

$\Rightarrow \frac{2a+12d}{-6d}=\frac{2a+6d}{-4d}$

$\Rightarrow \frac{a+6d}{-3}=\frac{a+3d}{-2}$

$\Rightarrow -2a-12d=-3a-9d$

$\Rightarrow a=3d$

Now, given that

$\Rightarrow k=\frac{a+3d}{a+9d}$

$\Rightarrow k=\frac{3d+3d}{3d+9d}$

$\Rightarrow k=\frac{6d}{12d}$

$\Rightarrow k=\frac{1}{2}$