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Hint: Assume Veena’s present age to be x and Kinjal’s present age to be y. Make two equations on x and y based on the information given in the question. Solve those two equations to find the values of x and y. The values of x and y will be the required present ages.

Complete step-by-step answer:

Let us assume that Veena’s present age is x years, and Kinjal’s present age is y years.

Based on the first condition, the ratio of these two ages is 2:3.

Thus, $\dfrac{x}{y}=\dfrac{2}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 1 \right)$

For the second condition, we know that since Veena’s present age is x, her age after 4 years will be $x+4$ years. Similarly, since Kinjal’s present age is y, age 4 years before would have been $y-4$years.

The ratio of these two ages is 4:1.

Hence, $\dfrac{x+4}{y-4}=\dfrac{4}{1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 2 \right)$

From (1), we obtain$x=\dfrac{2}{3}y$. Substituting this in equation (2), we get

$\begin{align}

& x+4=4\left( y-4 \right) \\

& \Rightarrow \dfrac{2}{3}y+4=4y-16 \\

& \Rightarrow \dfrac{2}{3}y=4y-20 \\

& \Rightarrow 4y-\dfrac{2}{3}y=20 \\

& \Rightarrow \dfrac{12y-2y}{3}=20 \\

& \Rightarrow \dfrac{10y}{3}=20 \\

\end{align}$

$\begin{align}

& \Rightarrow y=\dfrac{20\times 3}{10} \\

& \Rightarrow y=6 \\

\end{align}$

Thus, the present age of Kinjal is 6 years.

Now, to find the value of x, which gives the present age of Veena, we will substitute this value of y back in the expression obtained from (1), which is $x=\dfrac{2}{3}y$.

Thus $x=\dfrac{2}{3}\times 6$

$\Rightarrow x=4$

Thus the present age of Veena is 4 years.

Note: The solution obtained, the present ages of Veena and Kinjal can be verified by checking if these values satisfy the given conditions.

For the first condition, the ratio of their present ages is $x:y=4:6=2:3$. Thus the first condition is satisfied.

For the second condition, the ratio of Veena’s age after 4 years and Kinjal’s age 4 years before, $\left( x+4 \right):\left( y-4 \right)=8:2=4:1$. Hence, the second condition is also satisfied.

Thus, we have verified that the values of x and y obtained, 4 and 6 respectively is the correct answer.

Complete step-by-step answer:

Let us assume that Veena’s present age is x years, and Kinjal’s present age is y years.

Based on the first condition, the ratio of these two ages is 2:3.

Thus, $\dfrac{x}{y}=\dfrac{2}{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 1 \right)$

For the second condition, we know that since Veena’s present age is x, her age after 4 years will be $x+4$ years. Similarly, since Kinjal’s present age is y, age 4 years before would have been $y-4$years.

The ratio of these two ages is 4:1.

Hence, $\dfrac{x+4}{y-4}=\dfrac{4}{1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots \left( 2 \right)$

From (1), we obtain$x=\dfrac{2}{3}y$. Substituting this in equation (2), we get

$\begin{align}

& x+4=4\left( y-4 \right) \\

& \Rightarrow \dfrac{2}{3}y+4=4y-16 \\

& \Rightarrow \dfrac{2}{3}y=4y-20 \\

& \Rightarrow 4y-\dfrac{2}{3}y=20 \\

& \Rightarrow \dfrac{12y-2y}{3}=20 \\

& \Rightarrow \dfrac{10y}{3}=20 \\

\end{align}$

$\begin{align}

& \Rightarrow y=\dfrac{20\times 3}{10} \\

& \Rightarrow y=6 \\

\end{align}$

Thus, the present age of Kinjal is 6 years.

Now, to find the value of x, which gives the present age of Veena, we will substitute this value of y back in the expression obtained from (1), which is $x=\dfrac{2}{3}y$.

Thus $x=\dfrac{2}{3}\times 6$

$\Rightarrow x=4$

Thus the present age of Veena is 4 years.

Note: The solution obtained, the present ages of Veena and Kinjal can be verified by checking if these values satisfy the given conditions.

For the first condition, the ratio of their present ages is $x:y=4:6=2:3$. Thus the first condition is satisfied.

For the second condition, the ratio of Veena’s age after 4 years and Kinjal’s age 4 years before, $\left( x+4 \right):\left( y-4 \right)=8:2=4:1$. Hence, the second condition is also satisfied.

Thus, we have verified that the values of x and y obtained, 4 and 6 respectively is the correct answer.

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