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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?

Sagot :

MrRoyal

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]

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