Answer

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**Hint**: This is an Age based question and, in this question, we have to find the present ages of a mother and her daughter. First, we have to find a relation between the ages of the mother and the daughter then using that relation we can find the present ages of both of them.

**:**

__Complete step-by-step answer__Given:

Let us assume,

The present age of daughter Rekha is \[x\] years and the present age of mother Mrs. Goel is $y$ years.

Then according to the question,

The present age of mother Mrs. Goel = The present age of daughter Rekha + 27 years

So,

$\Rightarrow y = x + 27$

This is our first equation.

Now,

The age of mother Mrs. Goel after 8 years $ = \left( {y + 8} \right)$

And the age of daughter Rekha after 8 years $ = \left( {x + 8} \right)$

So, according to the question we have,

The age of mother Mrs. Goel after 8 years = $2 \times $( the age of daughter Rekha after 8 years)

$

\Rightarrow \left( {y + 8} \right) = 2 \times \left( {x + 8} \right)$

From first equation, substituting the value of $y$ in this equation we get,

$

\Rightarrow \left( {x + 27 + 8} \right) = 2x + 16\\

x + 35 = 2x + 16

$

Solving this we get,

$x = 19$

And substituting this value of $x$ in equation one we get,

$

y = 19 + 27\\

y = 46

$

Therefore, the present age of the daughter Rekha is 19 years and the present age of the mother Mrs. Goel is 46 years respectively.

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

**Note**: The alternate method of solving this question was to find two equations in terms of their ages which means in terms of $x$ and $y$ then solve those equations and find the value of their ages.

The equations we get are as following,

The first equation

$y = x + 27$

Or,

$x - y = - 27$

And similarly, the second equation

$

\left( {y + 8} \right) = 2 \times \left( {x + 8} \right)\\

y + 8 = 2x + 16\\

2x - y = - 8

$

From second equation – first equation we get,

$

2x - y - \left( {x - y} \right) = - 8 - \left( { - 27} \right)\\

x = 19

$

And substituting this value of $x$ in the first equation we get,

$

x - y = - 27\\

19 - y = - 27\\

y = 19 + 27\\

y = 46

$

Therefore, the age of daughter $ = 19{\rm{ years}}$

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