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Difficulty: MediumTranslating and Solving Algebraic Word Problems

A charity walkathon organizer pledges to donate a base amount of $150\$150 plus $2.50\$2.50 for every kilometer completed by each participant. On the day of the event, a participant completes a distance that is 33 kilometers less than twice their training average distance. If the organizer's donation for this participant is $212.50\$212.50, what is the participant's training average distance, in kilometers?

Answer: 14 kilometers

Answer

The participant's training average distance is 1414 kilometers.
The correct answer is 1414. To find this value, we write the donation relationship as 150+2.50d=212.50150 + 2.50d = 212.50, where dd is the distance completed. Solving for dd gives d=25d = 25 kilometers. Next, we translate the phrase '3 kilometers less than twice their training average distance' into the expression 2A−32A - 3, where AA is the training average. Equating this to the completed distance gives 2A−3=252A - 3 = 25. Solving for AA yields A=14A = 14.

Step-by-Step Solution

1
Set up an equation for the total donation as a function of the distance completed, dd, in kilometers.
150+2.50d=212.50150 + 2.50d = 212.50
The total donation is the sum of the flat base donation of $150\$150 and the rate of $2.50\$2.50 per kilometer completed.
2
Solve the equation for the completed distance, dd.
d=25d = 25
Subtracting 150150 from both sides of the equation yields 2.50d=62.502.50d = 62.50. Dividing both sides by 2.502.50 yields d=25d = 25.
3
Translate the relationship between the completed distance, dd, and the training average distance, AA, into an equation.
2A−3=252A - 3 = 25
The phrase '3 kilometers less than twice their training average distance' translates mathematically to 2A−32A - 3. Since the completed distance is 2525 kilometers, we set the expression equal to 2525.
4
Solve the equation for the training average distance, AA.
A=14A = 14
Adding 33 to both sides of the equation yields 2A=282A = 28. Dividing both sides by 22 yields A=14A = 14.

Key Concept

Translating word problems into multi-step linear equations and solving for the unknown variable.
Estimated Time:1m 30s
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Topics

Translating and Solving Algebraic Word Problems

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