Question

A is three times as old as B.C was twice as old as A four years ago. In four years’ time, A will be 31. What are the present ages of B and C?

Answer 1

We will first take the present ages of A,B and C as x,y and z respectively. We have been given different relationships between the ages of A and B and A and C. using these relations, we will form an equations in x and y and x and z. now, we have also been given the age of A after 4 four years of time. Subtracting 4 from that age, we will get the present age of A, i.e. x. Then we will put that value of x in the equations formed and calculate the required value of y and z.

Complete step by step answer:
Let the present ages of A,B and C be x,y and z respectively.
Now, we have been given that A is three times as old as B.
Thus, we can say that:
$x=3y$ …..(i)
Now, we have also been given that four years ago, C was twice as old as A.
Four years ago, age of A was the present age minus 4, i.e. $x-4$ and the age of C was also the present age minus 4, i.e. $z-4$ .
Thus we can say that:
$z-4=2\left( x-4 \right)$
By solving this equation, we will get a relationship between x and z.
Thus, we will get:
$\begin{aligned} & z-4=2\left( x-4 \right) \\\ & \Rightarrow z-4=2x-8 \\\ & \Rightarrow 8-4=2x-z \\\ \end{aligned}$
$\Rightarrow 2x-z=4$ ……(ii)
Now, we have been given that after four years, the age of A will be 31. After four years, the age of A will be his present age plus 4, i.e. $x+4$ .
Thus, we can say that:
$x+4=31$
By solving this equation, we will get the value of ‘x’. Thus, the value of ‘x’ is given as:
$\begin{aligned} & x+4=31 \\\ & \Rightarrow x=31-4 \\\ & \Rightarrow x=27 \\\ \end{aligned}$
Thus, the value of x is 27.
Now, we can put the value of x in equation (i) and equation (ii) and hence we will get our required value of y and z.
So, putting the value of ‘x’ in equation (i) we get:
$\begin{aligned} & x=3y \\\ & \Rightarrow 27=3y \\\ & \Rightarrow \dfrac{27}{3}=y \\\ & \Rightarrow y=9 \\\ \end{aligned}$
Thus, the present age of B is 9.
Now, putting the value of x in equation (ii) we get:
$\begin{aligned} & 2x-z=4 \\\ & \Rightarrow 2\left( 27 \right)-z=4 \\\ & \Rightarrow 54-z=4 \\\ & \Rightarrow 54-4=z \\\ & \Rightarrow z=50 \\\ \end{aligned}$
Thus, the present age of C is 50.

Thus, the present ages of B and C are 9 and 50 respectively.

Note: Take the relationships between the ages of A,B and C carefully. The relationship given can be confusing to some students as they some are relationship between present ages, some are between past and some between the future ages. So, this must be read carefully to obtain the right equations hence the correct answers.