Question

Three years hence, the ratio of thrice the age of
A and half of the age of B will be 4 : 1. Five years
ago, the age of A was half to the present age of B.
What is the present age of A?​

Answer 1

Topic :-

Linear Equation in Two Variables

Given :-

After three years, the ratio of thrice the age of A and half of the age of B will be 4 : 1.

Five years ago, the age of A was half to the present age of B.

To Find :-

Present age of A.

Solution :-

Let us assume that at present,

Age of A = x years

Age of B = y years

After three years,

Age of A = (x + 3) years

Age of B = (y + 3) years

It is given that,

After three years, the ratio of thrice the age of A and half of the age of B will be 4 : 1.

So after 3 years,

$\sf{\dfrac{Thrice\:the\:age\:of\:A}{Half\:the\:age\:of\:B}=\dfrac{4}{1}}$

$\sf{\dfrac{3(x+3)}{\dfrac{(y+3)}{2}}=\dfrac{4}{1}}$

$\sf{\dfrac{6(x+3)}{(y+3)}=\dfrac{4}{1}}$

Cross Multiply,

$\sf{6(x+3)=4(y+3)}$

$\sf{6x+18=4y+12}$

$\sf{4y-6x=18-12}$

$\sf{4y-6x=6}$

Dividing by 2 from both sides,

$\sf{\boxed{2y-3x=3}\longrightarrow Equation\:(1)}$

Five years ago,

Age of A = (x - 5) years

Age of B = (y - 5) years

It is given that,

Five years ago, the age of A was half to the present age of B.

So 5 years ago,

$\sf{x-5=\dfrac{y}{2}}$

Cross Multiplying,

$\sf{\boxed{2x-10=y}\longrightarrow Equation\:(2)}$

Substitute value of 'y' from Equation (2) in Equation (1),

$\sf{2y - 3x = 3}$

$\sf {2(2x - 10) - 3x = 3}$

$\sf{4x - 20 - 3x = 3}$

$\sf{4x - 3x = 20 + 3}$

$\sf{x = 23}$

Answer :-

The present age of A = x years = 23 years.