‘a’ and ‘b’ are two different numbers taken from the numbers $1 - 50.$ What is the largest value that $\dfrac{{a - b}}{{a + b}}$ can have? What is the largest value that $\left( {\dfrac{{a + b}}{{a - b}}} \right)$ can have?

Hint: Here we will assume the largest and the smallest values for the terms “a” and “b” to get the largest value for the given two expressions. Here for both expressions we will assume two different values and then will simplify placing the values in the expression.

Complete step by step answer:
For the largest value for $\dfrac{{a - b}}{{a + b}}$
Let us suppose that $a = 50$and $b = 1$
Place the values in the given expression –
$\dfrac{{a - b}}{{a + b}} = \dfrac{{50 - 1}}{{50 + 1}}$
Simplify the above expression finding the difference and the sum of the terms in the numerator and the denominator.
$\dfrac{{a - b}}{{a + b}} = \dfrac{{49}}{{51}}$ …. (A)
Now, similarly for the largest value for $\dfrac{{a + b}}{{a - b}}$.
Let us suppose that the value of $a = 50$ and $b = 49$
Place the given values in the above expression –
$\dfrac{{a + b}}{{a - b}} = \dfrac{{50 + 49}}{{50 - 49}}$
Simplify the above expression finding the sum and the difference of the terms in the above expression –
$\dfrac{{a + b}}{{a - b}} = \dfrac{{99}}{1}$
When there is one in the denominator term can be expressed as only the integer or the numerator part.
$\dfrac{{a + b}}{{a - b}} = 99$ …. (B)
Equations (A) and (B) are the required solution.

Note: Be careful about the sign convention and the expression given to assume the values from the given range. When there is difference in the numerator assume max value for “a” and min value for “b” and when there is difference in the denominator then assume max value for “a” and second next value for “b” as the smaller the denominator, larger the resultant value.