Question

If $(a-b):(a+b)=1:11$ , find the ratio $(5a+4b+15):(5a-4b+3)$ .

Answer 1

Step $1$: Find the relation between $a$ and $b$.

Given: $(a-b):(a+b)=1:11$

Taking $b$common from L.H.S. :

$\Rightarrow$ $\frac{b\left({\displaystyle \frac{a}{b}}-1\right)}{b\left({\displaystyle \frac{a}{b}}+1\right)}=\frac{1}{11}$

$\Rightarrow$ $11\left(\frac{a}{b}-1\right)=\left(\frac{a}{b}+1\right)$

$\Rightarrow$ $11\frac{a}{b}-11=\frac{a}{b}+1$

$\Rightarrow$ $10\frac{a}{b}=12$

$\Rightarrow$ $\frac{a}{b}=\frac{6}{5}$

$\Rightarrow$ $5a=6b$ or $a:b=6:5$

Step $2$ : Find the value of $(5a+4b+15):(5a-4b+3)$.

Let us assume $a=6k$, $b=5k$

Substituting the values in $(5a+4b+15):(5a-4b+3)$ we get

$$\begin{gathered} =\frac{5\left(6k\right)+4\left(5k\right)+15}{5\left(6k\right)-4\left(5k\right)+3} \\ =\frac{30k+20k+15}{30k-20k+3} \\ =\frac{50k+15}{10k+3} \\ =\frac{5(10k+3)}{\left(10k+3\right)} \\ =\frac{5}{1} \end{gathered}$$

Hence $(5a+4b+15):(5a-4b+3)=5:1$.