Question

Vandhana is four times as old as her brother Akash at present. After 10 years, she will be twice the age of her brother. Find their present ages.

Answer 1

Hint: In this question we should know that this question is solved by linear equations, at first we assume the values, then we write the given things in terms of assumed values then we get our first equation. Then we add the given value with the assumed one. After that we solve the equation then we substitute the first equation in the second equation (solved equation). Then we get our values.

Complete step by step answer:

Let us assume that the present age of Akash will be ‘x’ years
And the present ages of Vandhana will be ‘y’ years
And it is given that Vandhana is four times as old as her brother Akash
Therefore, according to condition
$y=4x$ … (1)
Now, after ten years
Akash’s age will be ‘x+10’ years
And, Vandhana’s age will be ‘y+10’ years
Again, it is given in the question that After 10 years, Vandhana will be twice the age of her brother ‘Akash’. Thus according to this condition,
 $
  2\left( {x + 10} \right) = y + 10 \\
   \Rightarrow 2x + 20 = y + 10 \\
 $
$\Rightarrow$ $2x + 20 - y = 10$ (Because transpose ‘y’ from RHS to LHS)
$ \Rightarrow 2x - y = 10 - 20$ (Because transpose ‘20’ from LHS to RHS)
$ \Rightarrow 2x - y = - 10$-----(2)
Now, we will put the value of ‘y’ from equation (1) in equation (2), we get
i.e., $
  2x - y = - 10 \\
   \Rightarrow 2x - 4x = - 10 \\
   \Rightarrow - 2x = - 10{\text{ }} \\
 $
$ \Rightarrow x = \dfrac{{10}}{2}$(‘-’ sign cancels and transposing ‘2’ from LHS to RHS)
$\Rightarrow$ $x = 5$
Therefore, the present age of Akash is ‘x’ i.e., 5 years

And the present age of her sister Vandhana is ‘y=4x’ i.e., $4 \times 5 = 20$ years.

Note: This question can also be solved by another method of eliminating the values like this,
According to condition in the question y = 4x or,
$y - 4x = 0$ …. (1)
Then After 10 years, Vandhana will be twice the age of her brother ‘Akash’. Thus according to this condition,
$
  2\left( {x + 10} \right) = y + 10 \\
   \Rightarrow 2x + 20 = y + 10 \\
 $
$ \Rightarrow 2x + 20 - y = 10$ (Because transpose ‘y’ from RHS to LHS)
$ \Rightarrow 2x - y = 10 - 20$ (Because transpose ‘y’ from LHS to RHS)
$ \Rightarrow 2x - y = - 10$----(2)
Now we have to multiply the values in order to make the variable in (1) and (2) equations the same.
So multiplying (1) by 2 and (2) by 2 we get,
$2y - 8x = 0$ …. (3) and $ - 2y + 4x = - 10$…. (4) . Now, adding (4) by (3) and rearranging we get,
$
  2y - 8x = 0 \\
  \underline { - 2y + 4x = - 20 + {\text{ }}} \\
  0 - 4x = - 20 \\
 $
Now, using the equation to find the value of x
$
   \Rightarrow x = \dfrac{{20}}{4} \\
   \Rightarrow x = 5 \\
 $
Now putting values in (1) we get,
$
  y - 4\left( 5 \right) = 0 \\
  y - 20 = 0 \\
  y = 20 \\
 $
By this method too you can accurately find values of ‘x’ (Akash) and ‘y’ (Vandhana) .(Also always remember that while subtracting operators in the equation mentioned below are changed but in this question we have added the equations so operators mentioned below are not changed).