GATE ECE 2024 | GA | Question: 3

Question

​​​​Five years ago, the ratio of Aman's age to his father's age was $1: 4$, and five years from now, the ratio will be $2:5$. What was his father's age when Aman was born?

• A. $28$ years
• B. $30$ years
• C. $35$ years
• D. $32$ years

Answer 1

Let aman's age = x years
Let aman's father age = y years
 
According to question:
Equation 1:
(x - 5)/(y - 5) = 1/4
4x - y = 15

Equation 2:
(x + 5)/(y + 5) = 2/5
5x - 2y = -15

Solving these 2 equations we get:
x = 15
y = 45
So, Aman's current age is 15 years
and Aman's father's current age is 45 years

The difference between their age is 30 years.

So when aman was born (his age will be 0) and his father's age would be 30 years old

So the answer to this question is Option (B)

Answer 2

$\text{Five years ago from now the age ratio was 1:4}$

$\text{After five years from now ratio is 2:5}$

$\text{timeline gap is 10 years}$

 

$\text{five year back_______now______after five years}$

$\text{$\frac{x + 10}{4x+ 10} = \frac{2}{5}$; their age ratio should be same}$

$\text{on solving this we get x = 10}$

 

$\text{The age of Aman was 10 and the age of his father was 40 from now.}$

$\text{So, His father was 40-10 = 30 year old when Aman was born(Age 0 years)}$

Answer 3

Let the present age of Aman and his father be $x$ and $y$ respectively.

Now, $\dfrac{x-5}{y-5} = \dfrac{1}{4}$

$\Rightarrow 4x - 20 = y-5$

$\Rightarrow 4x - y  = 15 \quad \longrightarrow (1)$

And. $\dfrac{x+5}{y+5} = \dfrac{2}{5}$

$\Rightarrow 5x+25 = 2y + 10$

$\Rightarrow 5x-2y = -15 \quad \longrightarrow (2)$

Multiply $2$ both sides on equation  $(1),$ we get

$\Rightarrow 8x - 2y  = 30\quad \longrightarrow (3)$

From equation $(2)$ and $(3)$

$5x-2y = -15 \quad \longrightarrow (2)$

$8x - 2y  = 30\quad \longrightarrow (3)$
____________________________
$-3x = -45$

$\Rightarrow x = 15$ years

And $y = 45$ years

$\therefore$ The age of his father's  when Aman was born  $ = 45-15 = 30$ years.

Correct Answer: B