Question

$A$ is twice as old as $B$. Ten years ago, $A$ was four times as old as $B$. What are their present ages?
A. $24$and$12$
B. $30$and$15$
C. $32$and$16$
D. $20$and$10$

Answer 1

Hint: Here in this question we have two conditions in which age of $B$ is dependent on the age of $A$. So, we can form two equations from these two conditions by which we will find the values of their ages.

Complete step-by-step answer: Here in this first of we will write the given information then we will start solving this question.
So, it is given in the question that
Now, $A$ is twice as old as $B$
Ten years ago, $A$ was four times as old as $B$.
So, as we have two conditions from which we can form two equations, so first of all let us assume that the present age of $B$ is $x$ years.
So, according to the first statement, that is $A$ is twice as old as $B$.
We can say that, Age of $A = 2 \times $(Age of $B$)
Age of $A = 2x$ …….(i)
Now, from the second statement that is, Ten years ago, $A$ was four times as old as $B$.
As ten years ago, Age of $B = x - 10$ and Age of $A = 2x - 10$
So, according to the statement Ten years ago, $A$ was four times as old as $B$.
$ \Rightarrow $ Age of $A = 4 \times $(Age of $B$)
So, by putting corresponding values we get
$ \Rightarrow $ $2x - 10 = 4 \times \left( {x - 10} \right)$
$ \Rightarrow $ $2x - 10 = 4x - 40$
$
   \Rightarrow 2x - 4x = - 40 + 10 \\
   \Rightarrow - 2x = - 30 \\
   \Rightarrow 2x = 30 \\
   \Rightarrow x = 15 \\
 $
Therefore,
Age of $B = 15$
And from equation (i) we can say that
Age of $A = 2 \times 15$
Age of $A = 30$

So, the correct answer is “Option B”.

Note: Here in this question we just required to form equations from the given conditions. The trick for the formation of such questions is that we will fix a variable according to our preference and according to that we will form the equations like here we formed the equations from fixing the age of $B$.