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Difficulty: Very hardLinear Equations in One and Two Variables

A courier service calculates its total delivery charge using a fixed base fee plus a constant per-mile rate. On Monday, a delivery of 15 miles received a 20% discount on the fixed base fee and a 25% surcharge on the per-mile rate, resulting in a total charge of 30.50.OnTuesday,adeliveryof20milesincurreda4030.50. On Tuesday, a delivery of 20 miles incurred a 40% increase on the fixed base fee and received a 15% discount on the per-mile rate, resulting in a total charge of 34.40. What is the standard total delivery charge, in dollars, for a 25-mile delivery with no fee adjustments or rate changes?

Answer: 40 dollars

Answer

The standard total delivery charge for a 25-mile delivery is 40 dollars.
Translating the scenario into linear equations gives 0.80B+18.75r=30.500.80B + 18.75r = 30.50 and 1.40B+17.00r=34.401.40B + 17.00r = 34.40. Solving this linear system yields a standard base fee of B=10.00B = 10.00 dollars and a standard per-mile rate of r=1.20r = 1.20 dollars. Substituting these into the standard 25-mile cost expression B+25rB + 25r yields 10.00+25(1.20)=40.0010.00 + 25(1.20) = 40.00 dollars.

Step-by-Step Solution

1
Formulate linear equations from the word problem context.
System of equations: 0.80B+18.75r=30.500.80B + 18.75r = 30.50 and 1.40B+17.00r=34.401.40B + 17.00r = 34.40.
Applying the percentage adjustments to the fixed base fee BB and the rate per mile rr for the given distances yields exact linear expressions.
2
Eliminate variable BB to solve for rr.
Multiply equations to equate coefficients of BB: 5.60B+131.25r=213.505.60B + 131.25r = 213.50 and 5.60B+68.00r=137.605.60B + 68.00r = 137.60. Subtracting gives 63.25r=75.9063.25r = 75.90, so r=1.20r = 1.20.
Finding the per-mile rate rr allows determination of the standard mileage component.
3
Solve for base fee BB using r=1.20r = 1.20.
1.40B+17(1.20)=34.40  ⟹  1.40B=14.00  ⟹  B=10.001.40B + 17(1.20) = 34.40 \implies 1.40B = 14.00 \implies B = 10.00.
Substituting rr into either linear equation gives the fixed base fee.
4
Calculate the target standard cost for 25 miles.
B+25r=10.00+25(1.20)=40.00B + 25r = 10.00 + 25(1.20) = 40.00.
Evaluating the standard pricing expression B+25rB + 25r with B=10B = 10 and r=1.20r = 1.20 gives the total cost.

Key Concept

Solving Systems of Two-Variable Linear Equations from Word Problems
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Topics

Linear Equations in One and Two Variables

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