Question

Hao Sheng, his father and his grandfather have an average age of 53. One-half of his grandfather's age, plus one-third of his father's age, plus one-fourth of Hao Sheng's age is 65. If 4 years ago, Hao Sheng's grandfather was four times as old as Hao Sheng, then how old are they now?

Answer 1

`x = 24` ---- Hao Sheng

`y =51` --- His father

`z = 84` ---- His grandfather

Explanation:

Given

`x \to Hao\ Sheng`

`y \to his\ father`

`z \to his\ grandfather`

From the question, we have:

`(x + y + z)/(3) = 53` --- The average of their age

`(z)/(2) + (y)/(3) +(x)/(4)= 65`

`z - 4 = 4(x - 4)` --- 4 years ago

Required

Their current age

`z - 4 = 4(x - 4)`

Open bracket

`z -4 = 4x-16`

Make z the subject

`z = 4x-16+4`

`z = 4x-12`

Multiply
`(z)/(2) + (y)/(3) +(x)/(4)= 65` by 12

`6z + 4y + 3x = 780`

Substitute
`z = 4x-12` in
`6z + 4y + 3x = 780`

`6(4x - 12) + 4y + 3x = 780`

`24x - 72 + 4y + 3x = 780`

Collect like terms

`24x + 3x + 4y = 780+ 72`

`27x + 4y = 852`

Substitute
`z = 4x-12` in
`(x + y + z)/(3) = 53`

`(x + y + 4x - 12)/(3) = 53\\`

Multiply through by 3

`x + y + 4x - 12 = 159`

Collect like terms

`x + 4x + y = 159+12`

`5x + y = 171`

Make y the subject

`y = 171 - 5x`

Substitute
`y = 171 - 5x` in
`27x + 4y = 852`

`27x + 4(171 - 5x) = 852`

`27x + 684 - 20x = 852`

Collect like terms

`27x - 20x = 852-684`

`7x = 168`

Divide by 7

`x = 24`

Solve for y in
`y = 171 - 5x`

`y =171 -5*24`

`y =171 -120`

`y =51`

Solve for z in
`z = 4x-12`

`z = 4 * 24- 12`

`z = 96- 12`

`z = 84`