Question

A dry cleaner has a computer program to determine the price it will charge an individual customer to clean a bag full of shirts (S) and pants (P). The total cost in dollars (C) is given by the following expression: $$C=10S+6P+5$$ What does the constant 5 most likely represent in the above expression? (A) A set fee the cleaner assesses to do any amount of cleaning (B) The cost to clean a shirt (C) The cost to clean a pair of pants (D) The total minimum cost to clean either one shirt or one pair of pants

Short answer

The constant 5 represents a set fee for any cleaning, hence option (A).

Step-by-step solution

Identify Components of Cost Expression

The cost expression provided is $C=10S+6P+5$. This expression calculates the total cost $C$ as the sum of three components: $10S$, $6P$, and $5$. Here, $10S$ represents the cost to clean $S$ shirts, and $6P$ represents the cost to clean $P$ pants.

Analyze Coefficients and Constant

In the expression, $10$ and $6$ are coefficients indicating the cost per unit for shirts and pants, respectively. The constant $5$ does not multiply by a variable and instead appears to be a fixed charge added to the computation of total cost.

Determine the Role of the Constant

The constant $5$ is not dependent on the number of shirts or pants, meaning it is applied to every customer's bill regardless of the number of items cleaned. Typically, such a constant represents a fixed fee added to the calculation, common in service industries as an overhead or processing fee.

Select the Most Likely Interpretation

Given the analysis, the constant $5$ is best interpreted as a set fee charged on top of the costs per item, which aligns with option (A): A set fee the cleaner assesses to do any amount of cleaning.

Key concepts

Mathematical Problem Solving

Mathematical problem solving involves breaking down a problem into smaller, manageable parts. This systematic approach helps in comprehending complex expressions and finding solutions. The equation given, $C=10S+6P+5$, is a classic example of applying this technique.
First, we identify the purpose of each element in the equation. Each term represents a part of the cleaning cost attributed to the number of shirts and pants cleaned and the fixed fee.

By dissecting the equation, you can notice:

• $10S$ accounts for the price to clean each shirt. Multiplying with $S$, the number of shirts, gives the total shirt cleaning cost.
• $6P$ is the analogous part for pants; $P$ is the variable here indicating quantity.
• The $5$ is a critical constant factor, which is independent of $S$ or $P$.

Understanding such breakdowns equips you to solve not only algebraic expressions but also practical problems encountered in everyday life, enhancing both analytical skills and critical thinking capabilities.

Interpretation of Constants

In algebraic expressions, constants play a vital role. They are the fixed numbers that do not change, no matter how the variables alter. In the given equation $C=10S+6P+5$, the constant $5$ is crucial and distinct from coefficients since it doesn't relate directly to shirts or pants.
When interpreting this constant, think about what remains constant in a real-world context. Here, the number 5 isn't contingent on the quantities $S$ or $P$. It's essentially like a baseline fee that charges even if no items are cleaned.

In many service-related industries, such fees cover costs unrelated to the quantity of goods processed, like administrative fees or other overhead expenses. Therefore, considering options provided, it makes sense to interpret this as (A) a set fee.
This understanding emphasizes the importance of recognizing constants in equations, which often represent implicit realities or constraints that aren't immediately obvious from the surrounding variables.

Linear Equations

Linear equations are straightforward mathematical models used to describe relationships where the variables appear with a power of one. The equation $C=10S+6P+5$ is a linear equation as neither $S$ (shirts) nor $P$ (pants) are raised to a higher power.
Each term in a linear equation can usually be explained as part of a simple addition or subtraction without any multiplication of variables themselves against each other.

Let's delve into the components:

• The term $10S$ adds up based on the number of shirts due to the direct multiplication.
• Similarly, $6P$ increases linearly with additional pants, showcasing a direct proportional relationship.
• The constant term, 5, shifts the entire line up the cost scale, serving as a flat fee regardless of the items.

Such linear equations allow for easy prediction and calculation of outcomes, essential for budgeting or logistical planning. Understanding this type of relationship is a cornerstone of algebraic education, vital for problem-solving in various fields, from economic forecasting to engineering.