Answer
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Hint: We will first let the present ages of mother and daughter be $x$ and $y$. Then, form the equations corresponding to the given conditions. We will solve both the equations to find the present ages of mother and daughter.
Complete step by step Answer:
First, let the present age of the mother be $x$ and the present age of the daughter be $y$
Now, we are given that the sum of the present ages of the mother and the daughter is 50 years.
Therefore, we can write it as, $x + y = 50$ (1)
The next condition is after 20 years.
But after 20 years, the age of the mother will be $x + 20$, and the age of the daughter will be $y + 20$
Now, we are given that the mother’s age will be twice her daughter’s age.
That is, \[x + 20\] will be twice of $y + 20$
We can write it as, $x + 20 = 2\left( {y + 20} \right)$
On solving it further, we will get
$x + 20 = 2y + 40$
On rearranging the equation and we will get $x - 2y = 20$ (2)
We will subtract the equation (1) and (2) to eliminate the terms containing $x$ and hence, find the value of $y$
$
x + y - x + 2y = 50 - 20 \\
3y = 30 \\
$
On dividing the equation throughout by 3, we will get,
$y = 10$
Now, substitute the value of $y$ in equation (1) to find the value of $x$
Therefore,
$
x + 10 = 50 \\
\Rightarrow x = 50 - 10 \\
\Rightarrow x = 40 \\
$
Hence, the present age of the mother is 40 years and the present age of the daughter is 10 years.
Note: Here, we have first calculated the value of $y$ and then by using substitution, we have calculated the value of $x$. But, we can also solve the question by first calculating the value of $x$ and then the value of $y$. Also, we can solve equations in two variables using substitution, elimination, and cross-multiplication method.
Complete step by step Answer:
First, let the present age of the mother be $x$ and the present age of the daughter be $y$
Now, we are given that the sum of the present ages of the mother and the daughter is 50 years.
Therefore, we can write it as, $x + y = 50$ (1)
The next condition is after 20 years.
But after 20 years, the age of the mother will be $x + 20$, and the age of the daughter will be $y + 20$
Now, we are given that the mother’s age will be twice her daughter’s age.
That is, \[x + 20\] will be twice of $y + 20$
We can write it as, $x + 20 = 2\left( {y + 20} \right)$
On solving it further, we will get
$x + 20 = 2y + 40$
On rearranging the equation and we will get $x - 2y = 20$ (2)
We will subtract the equation (1) and (2) to eliminate the terms containing $x$ and hence, find the value of $y$
$
x + y - x + 2y = 50 - 20 \\
3y = 30 \\
$
On dividing the equation throughout by 3, we will get,
$y = 10$
Now, substitute the value of $y$ in equation (1) to find the value of $x$
Therefore,
$
x + 10 = 50 \\
\Rightarrow x = 50 - 10 \\
\Rightarrow x = 40 \\
$
Hence, the present age of the mother is 40 years and the present age of the daughter is 10 years.
Note: Here, we have first calculated the value of $y$ and then by using substitution, we have calculated the value of $x$. But, we can also solve the question by first calculating the value of $x$ and then the value of $y$. Also, we can solve equations in two variables using substitution, elimination, and cross-multiplication method.
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