High School

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On a plane, both of you have a total of 135 kg of luggage. You paid $12 for your excess luggage, and your friend paid $24 for his excess luggage. Had all the luggage belonged to you, the excess luggage charge would have been $72.

At most, how many kilograms of luggage is each person permitted to bring on the plane free of additional charge?

Answer :

To solve this problem, we need to determine the maximum allowance of luggage, in kilograms, that each person is permitted to bring on a plane without incurring additional charges.

Let's break it down step by step:


  1. Define Variables:


    • Let [tex]x[/tex] be the maximum luggage allowance (in kg) per person without extra charge.

    • Let [tex]a[/tex] be the amount of luggage you have, and [tex]b[/tex] be the amount of luggage your friend has.

    • You paid $12 for your excess luggage, and your friend paid $24 for his excess luggage.



  2. Equations Based on Excess Charges:


    • If you had [tex]a[/tex] kg of luggage, the excess you would pay for would be [tex]a - x[/tex]. You paid $12, so:
      [tex]12 = k(a - x)[/tex]

    • Your friend with [tex]b[/tex] kg of luggage paid $24 for his excess, so:
      [tex]24 = k(b - x)[/tex]
      where [tex]k[/tex] is the charge per kg for excess luggage.



  3. Combined Luggage Charge:


    • If all the luggage belonged to you, you would have [tex]a + b[/tex] kg and pay $72 in total for the excess:
      [tex]72 = k((a + b) - x)[/tex]



  4. System of Equations:


    • From the equations, we identify:
      [tex]12 = k(a - x)[/tex]
      [tex]24 = k(b - x)[/tex]
      [tex]72 = k((a + b) - x)[/tex]



  5. Solving the Equations:


    • Dividing each side of the equations by [tex]k[/tex] gives:

      • [tex]a - x = \frac{12}{k}[/tex]

      • [tex]b - x = \frac{24}{k}[/tex]

      • [tex](a + b) - x = \frac{72}{k}[/tex]





  6. Substitute and Simplify:


    • From the above, it implies:
      [tex]\frac{12}{k} + \frac{24}{k} = \frac{72}{k}[/tex]

    • Simplifying gives us:
      [tex]36 = 72 \Rightarrow k = 2[/tex]

    • Substitute [tex]k[/tex] back:
      [tex]a - x = 6 \quad \text{(from \( a - x = \frac{12}{2} \))}[/tex]
      [tex]b - x = 12 \quad \text{(from \( b - x = \frac{24}{2} \))}[/tex]
      [tex](a + b) - x = 36 \quad \text{(from \( (a + b) - x = \frac{72}{2} \))}[/tex]



  7. Solving for Allowance [tex]x[/tex]:


    • By solving: [tex]x = a - 6[/tex] and [tex]x = b - 12[/tex], we find:
      [tex]a + b = 135[/tex]

    • Plug into equation for combined luggage:
      [tex](135) - x = 36 \Rightarrow x = 99[/tex]




Therefore, each person is permitted to bring at most 99 kg of luggage on the plane free of additional charge.

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Rewritten by : Barada