Answer

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**Hint:**The problem can be easily solved by using linear equations of variables. We can consider their ages as some particular variables and form equations according to the statement given. Further, we will find our solution by solving these equations.

**Complete step-by-step answer:**

Since the exact age of Pooja and her mother is not known,

Let the present age of Pooja $ = $ $x$years

Let the present age of her mother $ = $ $y$ years

By considering their ages as variables, we will be able to develop mathematical relationships between them. These equations and relationships will be created according to the statement of the problem.

It is given that at present,

Pooja is $21$years younger than her mother.

This can be mathematically expressed as,

$y$$ - $$x$$ = 21$

$y = 21 + x$ $ - - - - - - - - $$\left( 1 \right)$

Now another statement states that after $6$years from now, Pooja will be half her mother’s age.

Pooja’s age after $6$years $ = x + 6$

Her mother’s age after $6$years$ = y + 6$

But according to the statement,

$\left( {x + 6} \right)$is half of $\left( {y + 6} \right)$

$ \to $$x + 6 = \dfrac{{y + 6}}{2}$

$ \to x + 6 = \dfrac{y}{2} + 3$

$ \to y = 2x + 6$ $ - - - - - - - \left( 2 \right)$

Now we can solve these linear equations to calculate the present age of Pooja’s mother$\left( y \right)$,

$y = 21 + x$, $y = 2x + 6$

On subtracting both the above given equations, we get:

$y - y = 2x + 6 - \left( {21 + x} \right)$

$0 = 2x + 6 - 21 - x$

$0 = x - 15$

$x = 15$

Therefore, Pooja’s present age $ = 15$years.

Her mother’s present age $ = x + 21 = 15 + 21$

$ = 36$years.

**So, the correct answer is “Option A”.**

**Note:**The technique of converting the English statements and relations into mathematical equations or forms is vital to approach these kinds of problems. The statement should avoid ignoring any word while reading the statement of the question because it may affect the mathematical equations created.

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