Question

The sum of the present ages of the mother and daughter is 50 years. After 20 years, the mother’s age will be twice her daughter’s age at the time. Find their present ages.

Answer 1

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.