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Difficulty: MediumTranslating and Solving Algebraic Word Problems

A smart home heating system has two operating modes: Eco mode and Comfort mode. In a certain week, the system was active in one of these two modes for a total of 168168 hours. The number of hours the system was in Eco mode was 2424 hours less than three times the number of hours it was in Comfort mode. For how many hours was the heating system in Comfort mode during that week?

Answer: 48 hours

Answer

The heating system was in Comfort mode for 48 hours.
The correct answer is obtained by representing the hours in Comfort mode as cc and Eco mode as 3c−243c - 24. Since their sum is 168168, we write the equation c+3c−24=168c + 3c - 24 = 168. Solving this linear equation gives 4c=1924c = 192, which yields c=48c = 48.

Step-by-Step Solution

1
Define variables for the Comfort and Eco modes.
Let cc represent the hours in Comfort mode and ee represent the hours in Eco mode.
To represent the unknown quantities algebraically.
2
Translate the given information into a system of equations.
c+e=168c + e = 168 and e=3c−24e = 3c - 24
The sum of the hours in both modes is 168168, and the relationship between Eco and Comfort hours is described.
3
Substitute the expression for ee into the total hours equation.
c+(3c−24)=168c + (3c - 24) = 168
To create a single linear equation with one variable.
4
Solve for the variable cc.
4c−24=168⇒4c=192⇒c=484c - 24 = 168 \Rightarrow 4c = 192 \Rightarrow c = 48
Simplifying the equation gives the value of cc, which is the hours spent in Comfort mode.

Key Concept

Translating and Solving Algebraic Word Problems
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Topics

Translating and Solving Algebraic Word Problems

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