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Mrs. Goel is $27$ years older than her daughter Rekha. After $8$ years she will be twice as old as Rekha. Find their present ages.

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Last updated date: 13th Jun 2024
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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}}$