Question

Vani, her father and her grandfather have an average age of 53. One-half of her grandfather's age plus one-third of her father's age plus one-fourth of Vani's age is 65. Four years ago if Vani's grandfather was four times as old as Vani then what are the three equations we can use to find their present ages?

• A. $x+y+z=159$
• B. $6z+4y+3x=780$
• C. $6x+2y-z=12$
• D. $4x+0y-z=12$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $x+y+z=159$
B $6z+4y+3x=780$
D $4x+0y-z=12$

Let the present ages of Vani, her father and her grandfathers' be $x,yandz$ respectively.

Given: Average age = 53 .

$\Rightarrow \frac{x+y+z}{3}=53$
$\Rightarrow x+y+z=\mathrm{159............}\left(1\right)$

As given in the question, we can write as :

$\frac{z}{2}+\frac{y}{3}+\frac{x}{4}=65$

$\Rightarrow \frac{12z+8y+6x}{24}=65$

$\Rightarrow 12z+8y+6x=1560$

$\Rightarrow 6z+4y+3x=\mathrm{780........}\left(2\right)$

It is also given that " Four years ago ,Vani's grandfather was four times as old as Vani "

$\Rightarrow (z-4)=4(x-4)$

$\Rightarrow (z-4)=4x-16$

$\Rightarrow 4x-z-12=0$

$\Rightarrow 4x+0y-z=\mathrm{12.........}\left(3\right)$
​​​​​​​
​​​​​​​Now, we have

$x+y+z=\mathrm{159............}\left(1\right)$
$6z+4y+3x=\mathrm{780........}\left(2\right)$
​​​​​​​$4x+0y-z=\mathrm{12.........}\left(3\right)$