CHALLENGE If $\frac{a+1}{b-1}=\frac{5}{1}$and $\frac{a-1}{b+1}=\frac{1}{1},$find the value of $\frac{b}{a}$(Hint: Choose different values of and for which the proportions are true and evaluate the expression $\frac{b}{a}$.)

A ratio is a number, which expresses one quantity as a fraction of the other.

For example, the ratio of 3 to 4 is 3 : 4 or $\frac{3}{4}$.

According to the hint we can choose any number to see if it fits the ratio and make it true. Let us choose the most basic numbers first like set$\left(a,b\right)$ as$\left(1,1\right)\left(1,2\right),\left(2,1\right)\left(2,2\right)$ and so on…

We are given the equations $\frac{a+1}{b-1}=\frac{5}{1}$and$\frac{a-1}{b+1}=\frac{1}{1}$

We start by plugging the first value: $a=1,b=1$

Thus we have:

$\begin{array}{l}\frac{a+1}{b-1}=\frac{5}{1}\\ \frac{1+1}{1-1}=\frac{5}{1}\\ \frac{1}{0}\ne \frac{5}{1}\end{array}$

Thus $a=1,b=1$is not a valid solution.

Similarly plugging in next set $a=1,b=2$we get,

$\begin{array}{l}\frac{a+1}{b-1}=\frac{5}{1}\\ \frac{1+1}{2-1}=\frac{5}{1}\\ \frac{2}{1}\ne \frac{5}{1}\end{array}$

Thus $a=1,b=2$is not a valid solution…

Going in this manner we get to the solution set $a=4,b=2$

Plugging in this set $a=4,b=2$we get,

$\begin{array}{l}\frac{a+1}{b-1}=\frac{5}{1}\\ \frac{4+1}{2-1}=\frac{5}{1}\\ \frac{5}{1}=\frac{5}{1}\end{array}$

Thus $a=4,b=2$is a valid solution.

Plugging in the valid set $a=4,b=2$, in the second equation we get,

$\begin{array}{l}\frac{a-1}{b+1}=\frac{1}{1},\\ \frac{4-1}{2+1}=\frac{1}{1}\\ \frac{3}{3}=\frac{1}{1}\\ \frac{1}{1}=\frac{1}{1}\end{array}$

Thus $a=4,b=2$is indeed a valid solution.

Since $a=4,b=2$is a valid solution. We have,

$\begin{array}{l}\frac{b}{a}=\frac{2}{4}\\ =\frac{1}{2}\end{array}$