Respuesta :
Answer:
[tex]x = 24[/tex] ---- Hao Sheng
[tex]y =51[/tex] --- His father
[tex]z = 84[/tex] ---- His grandfather
Step-by-step explanation:
Given
[tex]x \to Hao\ Sheng[/tex]
[tex]y \to his\ father[/tex]
[tex]z \to his\ grandfather[/tex]
From the question, we have:
[tex]\frac{x + y + z}{3} = 53[/tex] --- The average of their age
[tex]\frac{z}{2} + \frac{y}{3} +\frac{x}{4}= 65[/tex]
[tex]z - 4 = 4(x - 4)[/tex] --- 4 years ago
Required
Their current age
[tex]z - 4 = 4(x - 4)[/tex]
Open bracket
[tex]z -4 = 4x-16[/tex]
Make z the subject
[tex]z = 4x-16+4[/tex]
[tex]z = 4x-12[/tex]
Multiply [tex]\frac{z}{2} + \frac{y}{3} +\frac{x}{4}= 65[/tex] by 12
[tex]6z + 4y + 3x = 780[/tex]
Substitute [tex]z = 4x-12[/tex] in [tex]6z + 4y + 3x = 780[/tex]
[tex]6(4x - 12) + 4y + 3x = 780[/tex]
[tex]24x - 72 + 4y + 3x = 780[/tex]
Collect like terms
[tex]24x + 3x + 4y = 780+ 72[/tex]
[tex]27x + 4y = 852[/tex]
Substitute [tex]z = 4x-12[/tex] in [tex]\frac{x + y + z}{3} = 53[/tex]
[tex]\frac{x + y + 4x - 12}{3} = 53\\[/tex]
Multiply through by 3
[tex]x + y + 4x - 12 = 159[/tex]
Collect like terms
[tex]x + 4x + y = 159+12[/tex]
[tex]5x + y = 171[/tex]
Make y the subject
[tex]y = 171 - 5x[/tex]
Substitute [tex]y = 171 - 5x[/tex] in [tex]27x + 4y = 852[/tex]
[tex]27x + 4(171 - 5x) = 852[/tex]
[tex]27x + 684 - 20x = 852[/tex]
Collect like terms
[tex]27x - 20x = 852-684[/tex]
[tex]7x = 168[/tex]
Divide by 7
[tex]x = 24[/tex]
Solve for y in [tex]y = 171 - 5x[/tex]
[tex]y =171 -5*24[/tex]
[tex]y =171 -120[/tex]
[tex]y =51[/tex]
Solve for z in [tex]z = 4x-12[/tex]
[tex]z = 4 * 24- 12[/tex]
[tex]z = 96- 12[/tex]
[tex]z = 84[/tex]