Problem 12 - Algebra and Trigonometry

Question

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The total rent paid for an apartment if the rent is \(\$ 795\) a month; \(n=\) number of months

Step-by-Step Solution

Answer

Total rent = \( 795n \).

Step 1: Understanding the Problem

We need to express the total rent paid as a function of the number of months, denoted by \( n \). Each month's rent costs \( \$795 \).

Step 2: Calculating Total Rent

To find the total rent paid for \( n \) months, we multiply the monthly rent by the number of months. Thus, the expression is \( 795 \times n \).

Step 3: Final Expression

Therefore, the total rent paid, in terms of \( n \), is expressed as \( 795n \).

Key Concepts

Linear Equation

A linear equation is a type of equation that showcases a direct relationship between two variables. The foundational structure of a linear equation is an expression that equates to a straight line when plotted on a graph. This is why it's called "linear."

In this exercise, we are examining the relationship between the total rent paid and the number of months the rent is paid for. The linear relationship here is represented by the equation: \[t = 795n\]where \( t \) is the total rent and \( n \) is the number of months. This equation illustrates how the total amount paid changes directly with the number of months. Every time \( n \) increases by one, the total amount \( t \) increases by \( \$795 \), forming a straight line if you were to draw it on a graph.

Variables in Expressions

Variables are symbols that are used to represent unknown or changeable quantities in mathematical expressions and equations. By using variables, we can construct expressions that easily model a wide range of scenarios.

In our exercise, the variable \( n \) is used to represent the number of months for which rent is paid. This variable allows us to calculate the total rent for any given number of months. For instance, if \( n = 3 \), then the expression becomes:\[795 \times 3 = 2385\]indicating that the total rent for three months is \( \$2385 \). With variables, you have the flexibility to substitute different values, which is particularly useful in drawing general conclusions from specific examples.

Arithmetic Operations

Arithmetic operations are the basic processes we use in mathematics to manipulate numbers. These include addition, subtraction, multiplication, and division. In this exercise, multiplication is the key operation used.

The problem requires us to determine the total rent over several months. To do this, we multiply the monthly rent \( \$795 \) by the number of months \( n \). This operation can be outlined in a general formula:\[\text{Total Rent} = \text{Monthly Rent} \times \text{Number of Months} = 795n\]This multiplication highlights how simple arithmetic operations can be applied to solve real-world problems and derive important solutions. Multiplication helps to scale a known quantity by a certain factor, which is essential for calculating totals, costs, or other cumulative metrics.