Problem 6 - Algebra and Trigonometry

Question

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

The total rent is \( 795n \).

Step 1: Understanding the Problem

We need to find an expression that represents the total rent paid over a period of months. The monthly rent is given as \( \$795 \) per month, and the number of months is represented by \( n \).

Step 2: Identifying Known and Unknown Variables

The known variable is the rent per month, which is \( 795 \). The unknown is the total rent paid, and the variable \( n \) represents the number of months.

Step 3: Applying the Multiplication Concept

The total rent paid is calculated by multiplying the monthly rent by the number of months the rent is paid. Thus, the total rent is \( 795 \times n \).

Step 4: Expressing the Quantity

Finally, the total rent paid, expressed in terms of \( n \), is given by the expression \( 795n \).

Key Concepts

Variables and Constants

In algebra, it's important to understand the difference between variables and constants. A variable is a symbol, like \( n \), that represents numbers that can change or vary. In our exercise, \( n \) is the variable, standing for the number of months. This is because the number of months can differ depending on the situation.
On the other hand, a constant is a fixed value that does not change. In our example, \( 795 \) is a constant because the rent per month is always $795. Constants are often used to represent quantities that remain the same across different scenarios. To put it simply:

• Variables: Changeable parts of an expression (e.g., the number of months \( n \)).
• Constants: Unchanging parts of an expression (e.g., the monthly rent \( 795 \)).

Multiplication in Algebra

When you tackle algebraic problems, multiplying variables and constants is a common task. This exercise asked us to find the total rent by multiplying the rent per month by the number of months, which means using multiplication. We calculate the total rent by multiplying the constant \( 795 \) and the variable \( n \). The result is the expression \( 795n \).
This demonstrates that simple arithmetic operations, like multiplication, are integrated into algebraic expressions. Notably, when multiplying a number by a variable, it's common to just write them side by side—no need for the multiplication sign. Thus, 795 times \( n \) is written as \( 795n \). This makes the expression neat and compact and is a standard practice in algebraic notation.

Solving Problems with Equations

Solving problems using equations is all about creating connections between known and unknown quantities. In the given problem, we created an algebraic expression \( 795n \) to represent the total rent over \( n \) months. This equation allows us to solve for the total cost if \( n \) is specified.
Having an equation or expression like \( 795n \) makes it easy to quickly calculate the total rent for any number of months by substituting the variable \( n \) with a specific number. For example:

• If \( n = 2 \), then the total rent equals \( 795 \times 2 = 1590 \).
• If \( n = 6 \), then the total rent equals \( 795 \times 6 = 4770 \).

By using equations, we can solve real-world problems efficiently, making life a little simpler. This allows us to understand how changing one element, like the number of months, affects the overall outcome.