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Difficulty: MediumTranslating and Solving Algebraic Word Problems

A local courier service offers two delivery options inside the city limits: Standard and Express. A Standard delivery costs a flat fee of 15.00plus15.00 plus 0.75 per mile. An Express delivery costs a flat fee of 25.00plus25.00 plus 0.50 per mile. If the cost of a Standard delivery is exactly $5.00 less than the cost of an Express delivery, what is the total cost of the Express delivery?

  1. A
    $30.00
  2. $35.00Answer
  3. C
    $45.00
  4. D
    $55.00
  5. E
    $60.00

Answer

The total cost of the Express delivery is $35.00.
The correct answer is 35.00.First,representthecostsasalgebraicfunctionsofthedistance35.00. First, represent the costs as algebraic functions of the distance m :: 15 + 0.75m forStandardand for Standard and 25 + 0.50m forExpress.SetuptheequationreflectingthattheStandardcostis for Express. Set up the equation reflecting that the Standard cost is 5.00 less than the Express cost: 15+0.75m=(25+0.50m)−515 + 0.75m = (25 + 0.50m) - 5. Simplifying this equation gives 15+0.75m=20+0.50m15 + 0.75m = 20 + 0.50m. Subtracting 0.50m0.50m and 1515 from both sides results in 0.25m=50.25m = 5, which yields m=20m = 20. Finally, plug m=20m = 20 into the Express cost function: 25+0.50(20)=35.0025 + 0.50(20) = 35.00 dollars.

Step-by-Step Solution

1
Define the variables and write the cost functions for both options.
Let mm represent the number of miles. The Standard delivery cost is S=15+0.75mS = 15 + 0.75m, and the Express delivery cost is E=25+0.50mE = 25 + 0.50m.
Translating the verbal descriptions into algebraic expressions establishes the basis for solving the problem.
2
Set up the equation using the given relationship that Standard is 5.00 dollars less than Express.
15+0.75m=(25+0.50m)−515 + 0.75m = (25 + 0.50m) - 5
The phrase 'Standard cost is 5.00 dollars less than Express cost' translates to S=E−5S = E - 5.
3
Solve the equation for mm.
15+0.75m=20+0.50m  ⟹  0.25m=5  ⟹  m=2015 + 0.75m = 20 + 0.50m \implies 0.25m = 5 \implies m = 20 miles.
Solving for mm determines the delivery distance that satisfies the cost relationship.
4
Calculate the total cost of the Express delivery by substituting the value of mm into the Express cost function.
E=25+0.50(20)=25+10=35E = 25 + 0.50(20) = 25 + 10 = 35 dollars.
The question specifically asks for the total cost of the Express delivery, not the distance or the Standard delivery cost.

Key Concept

Translating verbal descriptions of costs and differences into algebraic equations and solving them.

Alternative Method

You can test the choices. For example, if the Express cost is 35.00,thenthedistance35.00, then the distance m isfoundbysolving is found by solving 25 + 0.50m = 35 \implies 0.50m = 10 \implies m = 20 .At. At 20 miles,theStandardcostis miles, the Standard cost is 15 + 0.75(20) = 30 .Since. Since 30 isindeedexactly is indeed exactly 5 lessthan less than 35$, this choice is correct.
Estimated Time:1m 30s
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Topics

Translating and Solving Algebraic Word Problems

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