Answer

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**Hint:**We will consider Salil’s age be $x$ years and Sargams’s age $y$ years. Then from the given data we will form two equations and then we will solve them to get the value of $x$ and $y$. These will be the ages or we can say present ages of Salil and Sargam.

**Complete step by step solution:**

Let the present ages of Salil and Sargam be x and y years respectively.

The first condition is, Salil’s age is $23$ years more than half of Sargams’s age. This can be formed in the equation as,

\[x = \dfrac{y}{2} + 23\] ….equation (1)

taking the LCM on RHS,

\[x = \dfrac{{y + 23 \times 2}}{2}\]

On cross multiplying $2$ and taking the product on RHS,

\[2x = y + 46\]

Taking the variables on one side we get,

\[2x - y = 46\] this is simplified equation (1)

Now from the second condition, five years ago, the sum of their ages was $55$ years. This can be formed in the equation as,

\[\left( {x - 5} \right) + \left( {y - 5} \right) = 55\] …..equation (2)

On solving the brackets,

\[x + y - 10 = 55\]

Taking the constants on one side we get,

\[x + y = 55 + 10\]

\[\Rightarrow x + y = 65\] this is simplified equation (2)

On adding equation1 and equation (2),

\[2x - y + x + y = 46 + 65\]

Cancelling y and adding the x terms from the equation we get,

\[3x = 111\]

Now on dividing 111 by 3 we get,

\[x = \dfrac{{111}}{3}\]

\[ \Rightarrow x = 37\]

Now this is the present age of Salil. Now putting this value in any of the simplified equations we get,

\[ y = 65 - x \\

\Rightarrow y = 65 - 37 \\

\Rightarrow y = 28 \\

\]

This is the present age of Sargam.

**Therefore, the present ages of Salil and Sargam are 37 years and 28 years respectively.**

**Note:**

Note that forming the equation using the data or condition is the most important part of these problems. Because solution of the equation is the answer of the question.

Also note that some years before is subtracting this number from the present age and some years from now means adding that number in the present age.

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