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Difficulty: MediumLinear Equations in One Variable

A local logistics company charges a flat account setup fee of 55 dollars plus 44 dollars per pound for the first 66 pounds of a package shipment. For any weight beyond 66 pounds, the company charges 33 dollars per pound. If the total cost to ship a package is 5050 dollars, what is the total weight of the package, in pounds?

  1. A
    7
  2. B
    9
  3. 13Answer
  4. D
    1

Answer

The correct answer is 13, which represents the total weight of the package in pounds.
To find the total weight, we first determine the cost of the first 66 pounds, which is the 55 dollars flat fee plus 44 dollars per pound, totaling 2929 dollars. Subtracting this baseline cost from the total cost of 5050 dollars leaves 2121 dollars for the weight exceeding 66 pounds. Since the rate for the excess weight is 33 dollars per pound, the excess weight is 2121 divided by 33, which is 77 pounds. Adding the initial 66 pounds gives a total weight of 1313 pounds.

Step-by-Step Solution

1
Calculate the total cost of shipping the first 6 pounds of the package.
The cost for the first 66 pounds, including the flat setup fee, is 5+6×4=295 + 6 \times 4 = 29 dollars.
This establishes the baseline cost before the rate of 33 dollars per pound is applied to any weight exceeding 66 pounds.
2
Set up a linear equation representing the total shipping cost of 50 dollars for a package of weight w, where w is greater than 6.
The equation is 29+3(w−6)=5029 + 3(w - 6) = 50.
The total cost is the sum of the cost for the first 66 pounds (2929 dollars) and the cost of the excess weight (w−6w - 6) at the rate of 33 dollars per pound.
3
Solve the linear equation for w by isolating the variable.
3(w−6)=21  ⟹  w−6=7  ⟹  w=133(w - 6) = 21 \implies w - 6 = 7 \implies w = 13.
Subtracting 2929 from both sides and then dividing by 33 isolates the term representing the excess weight, and adding 66 yields the total weight.

Key Concept

Linear Equations in One Variable
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Topics

Linear Equations in One Variable

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