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 how old are they all now?

Answer 1

Let the age of Vani be ”x” years

Vani father age = “y” years

Vani grandfather age = “z” years

By the given first condition.

`(x + y + z)/3` = 53

x + y + z = 159 ….(1)

By the given 2^nd Condition.

`1/2  z + 1/3  y + 1/4  x` = 65

Multiply by 12

6z + 4y + 3x = 780

3x + 4y + 6z = 780 ….(2)

By the given 3^rd condition

z – 4 = 4 (x – 4) ⇒ z – 4 = 4x – 16

– 4x + z = – 16 + 4

4x – z = 12 ……(3)

(1) × 4 ⇒  4x + 4y + 4z = 636     ....(1)
(2) × 1 ⇒  3x + 4y + 6z = 780     ....(2)
            ⇒    (–)  (–)   (–)      (–)            
(1) – 2  ⇒   x + 0 – 2z = – 144
                        x – 2z = – 144      ....(4)
(3) × 2 ⇒       8x – 2z = 24           ....(3)
(4) × 1 ⇒         x – 2z = – 144      ....(4)
                       (–)  (+)   (+)                 
(3) – (4) ⇒      7x + 0 = 168
                              x = `168/7` = 24

Vani age = 24 years

Vani’s father age = 51 years

Vani grandfather age = 84 years

Substitute the value of x = 24 in (3)

4 (24) – z = 12

96 – z = 12

– z = 12 – 96

z = 84

Substitute the value of x = 24 and z = 84 in (1)

x + y + z = 159

24 + y + 84 = 159

y + 108 = 159

y = 159 – 108

= 51