Question

Sum of the ages of X and Y, 12 years ago, was
48 years and sum of the ages of X and Y, 12
years hence will be 96 years. Present age of X is?​

Answer 1

Given :

• Sum of the ages of X and Y, 12 years ago was 48 years.
• Sum of the ages of X and Y, 12 years hence will be 96 years.

To Find :

• Present age of X.

Solution :

Let the present age of X be a years.

Let the present age of Y be b years.

Case 1 :

Age of X and Y 12 years ago,

• X → (a-12) years
• Y → (b-12) years

Equation :

$\sf{(a-12) +(b-12) =48}$

$\sf{a-12+b-12=48}$

$\sf{a+b-12-12=48}$

$\sf{a+b-24=48}$

$\sf{a+b=48+24}$

$\sf{a+b=72\:\:(i)}$

Case 2 :

Age of X and Y 12 years hence,

• X → (a+12) years
• Y → (b+12) years.

Equation :

$\sf{(a+12)+(b+12)=96}$

$\sf{a+12+b+12=96}$

$\sf{a+b+12+12=96}$

$\sf{a+b+24=96}$

$\sf{a+b=96-24}$

$\sf{a+b=72\:\:(ii)}$

Equation (i) and (ii) are consistent linear equation in two variables.

Hence, determining the present age of X is not possible.

Answer 2

AnswEr :

$\bf{\green{\large{\underline{\underline{\bf{Given\::}}}}}}$

Sum of the ages of X and Y is 12 years ago, was 48 years and sum of the ages of X and Y is 12 years hence will be 96 years.

$\bf{\red{\large{\underline{\underline{\bf{To\:find\::}}}}}}$

Present age of X is

$\bf{\purple{\large{\underline{\underline{\bf{Explanation\::}}}}}}$

Let the present age of X be r years.

Let the present age of Y be m years.

$\underbrace{\bf{12\:years\:agO\::}}}}}$

$\leadsto\sf{The\:age\:of\:X\:is\:=\:(r-12)\:years.}}\\\\\leadsto\sf{The\:age\:of\:Y\:is\:=\:(m-12)\:years}}$

$\bf{\red{\underline{\underline{\tt{A.T.Q\::}}}}}$

$\implies\tt{(r-12)+(m-12)=48}\\\\\\\\\implies\tt{r-12+m-12=48}\\\\\\\\\implies\tt{r+m-24=48}\\\\\\\\\implies\tt{r+m=48+24}\\\\\\\\\implies\tt{\purple{r+m=72...............(1)}}$

______________________________________

$\underbrace{\bf{After\:12\:years\::}}}}}$

$\leadsto\sf{The\:age\:of\:X\:is\:=\:(r+12)\:years.}}\\\\\leadsto\sf{The\:age\:of\:Y\:is\:=\:(m+12)\:years}}$

$\bf{\red{\underline{\underline{\tt{A.T.Q\::}}}}}$

$\implies\tt{(r+12)+(m+12)=96}\\\\\\\\\implies\tt{r+12+m+12=96}\\\\\\\\\implies\tt{r+m+24=96}\\\\\\\\\implies\tt{r+m=96-24}\\\\\\\\\implies\tt{\purple{r+m=72...............(2)}}$

Here, both equations are same.

∴ The pair of linear equation is consistent.