Question

A is twice as old as B, ten year ago he was four times as old. What is their present age?

Answer 1

Hint: In such a type of question assume the present age of A and B as $x$ and $y$. Ten year ago, their ages were $x-10$and $y-10$ years respectively. Form the equation as per the given information in question. Try to make two independent equations in terms of $x$ and $y$. Form two equations in the form of \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\], solve the equation so obtained to get the values of unknowns.

Complete step-by-step answer:
Suppose the present age of A is $x$and that of B is $y$.
It is given that A is twice as old as B so we can write
$x=2y---(a)$
Now ten year ago we can write
Age of A = $x-10$
Age of B=$y-10$
From question ten year ago age of A was four times that of B, hence we can write
$x-10=4(y-10)-----(b)$
On simplifying the above equation, we can write
$\begin{align}
  & x-10=4y-40 \\
 & \Rightarrow x-4y=10-40 \\
 & \Rightarrow x-4y=-30-----(c) \\
\end{align}$
Now we get two linear equation in two variables that is equation $(a)$ and equation $(c)$
Here we have to solve the equation $(a)$and equation $(c)$so we substitute the value of $x=2y$
In equation $(c)$, so we can write
$\begin{align}
  & 2y-4y=-30 \\
 & \Rightarrow -2y=-30 \\
\end{align}$
Dividing both side by -2, we get
$y=15$
Now substituting the value of $y$in equation $(a)$ we get
$\begin{align}
  & x=2(15) \\
 & \Rightarrow x=30 \\
\end{align}$
Hence
Present Age of A = 30 years
Present Age of B = 15 years


Note: We can solve the above equation by elimination method also. Whether two linear equation have unique solution or not we check the ratio of coefficient of variables as,
If \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] are two linear equation then they have unique solution if $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$. Also we can draw the graph of equation $(a)$ and $(d)$ as the equations satisfy the above condition, both lines cut each other. The point of intersection is the solution of the equation on the $x$and $y$axis respectively.