Question

A grandfather is ten times older than his granddaughter. He is also 54 year older than her. Find their present ages.
A. The present age of granddaughter is 6 years and that of grandfather is 60 years.
B. The present age of granddaughter is 8 years and that of grandfather is 80 years.
C. The present age of granddaughter is 7 years and that of grandfather is 70 years.
D. The present age of granddaughter is 9 years and that of grandfather is 90 years.

Answer 1

Hint: In this question first try to write the given information into a linear equation of two variables then solve the simultaneous equations. If $ {{a}_{1}}x+{{b}_{1}}y+{{c}_{2}}=0 $ and $ {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0 $ are two linear equation and if $ \dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}} $ then both equation is satisfied by a unique value of (x,y) which is the solution of simultaneous equations. To solve this, find the value of one variable in terms of another and put the value so obtained in the second equation and solve it to get the solution.

Complete step-by-step answer:
Let us assume that the present age of grandfather is x and present age of granddaughter is y.
So from question as grandfather is ten times older than his granddaughter so we can write
 $ \Rightarrow x=10y-----(a) $
Also grandfather is $ 54 $ years older than her, so we can write
 $ \Rightarrow x-y=54----(b) $
Now substituting the value if x from equation (a) into equation (b)
 $ \begin{align}
  & \Rightarrow 10y-y=54 \\
 & \Rightarrow 9y=54 \\
\end{align} $
Diving both side by 9 we get
 $ \begin{align}
  & \Rightarrow \dfrac{9y}{9}=\dfrac{54}{9} \\
 & \Rightarrow y=6 \\
 & \\
\end{align} $
Now putting the value of y in equation (a) we get
 $ \begin{align}
  & x=10\times 6 \\
 & \Rightarrow x=60 \\
\end{align} $
Hence the present age of grandfather is 60 years and the present age of granddaughter is 6 years.
So option A is correct.

Note: We can solve the equations by Elimination method also. We must check the answer by putting it in the equation.
Geometrically, if $ {{a}_{1}}x+{{b}_{1}}y+{{c}_{2}}=0 $ and $ {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0 $ are two linear equation they represent two independent straight lines.
IF $ \dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}} $ both straight lines intersect each other. The point of intersection lies on both lines so they satisfy both equations of the straight line hence the point (x,y) which is the solution of simultaneous equations.