Problem 47 - Algebra 1

Question

Hotel RATES A hotel charges \( 49.99 \) per room per night for adults and \( 44.10\) per room per night for senior citizens. The expression \(2 \)x\( 49.99+ \)3 x 44.1 represents the total cost of five rooms for two adults and three senior citizens for an overnight stay. Where in the expression can you put grouping symbols to make sure it is evaluated correctly?

Step-by-Step Solution

Answer

The correct way to group the expression would be (2 x 49.99) + (3 x 44.1). The total cost for two adults and three senior citizens for an overnight stay in five rooms is $232.28.

Step 1: Understand the Expression

The expression 2 x 49.99 + 3 x 44.1 contains two parts: the first part (2 x 49.99) represents the cost for two adults' rooms, while the second part (3 x 44.1) depicts the cost for three seniors' rooms.

Step 2: Apply Grouping Symbols

To make sure the expression is evaluated correctly, it's best to add parentheses around each multiplication operation. This results in the expression: (2 x 49.99) + (3 x 44.1). This ensures that each multiplication operation will be done before adding the results together following the order of operations.

Step 3: Carry out the Operations

Now that the expression is properly grouped, carry out the multiplication operations within each parentheses, then add the two products together to get the total cost as follows: (2 x 49.99) + (3 x 44.1) equals to 99.98 + 132.3, which sums up to 232.28.

Key Concepts

Order of Operations

Understanding the order of operations is essential when working with algebraic expressions. It refers to the rules that determine the sequence in which parts of a mathematical expression should be evaluated to obtain the correct result. The standard order is often abbreviated as PEMDAS:

• Parentheses
• Exponents (including roots, such as square roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the example of the hotel charges, we calculate the cost for each room type separately, then sum them up. Multiplication (calculating the cost per room type) takes precedence over addition (summing up the different room types). Without proper attention to the order of operations, one might incorrectly add before multiplying, which would lead to an inaccurate total cost. This solidifies why the order of operations isn't just academic—it's crucial for everyday calculations.

Grouping Symbols

Grouping symbols like parentheses, brackets, and braces, are used in algebra to indicate which operations should be performed first in an expression. When an expression includes these, you do all the calculations inside the grouping symbols before doing anything outside. In our hotel rate calculation, we added parentheses around each multiplication operation:

\textbf{With Grouping:}
\((2 \times 49.99) + (3 \times 44.1)\)
\textbf{Without Grouping:}
\(2 \times 49.99 + 3 \times 44.1\).

Without these symbols, one might misinterpret the calculations required. By using parentheses, we clearly define the tasks: calculate the cost for two adults and three seniors separately and only then add these amounts together. This not only prevents mistakes but also helps others understand the steps you've taken in your calculations.

Real-World Algebra Application

Algebra is not confined to the classroom; it's a tool that we use in everyday life, often without even realizing it. In real-world contexts, algebra helps us make sense of situations that involve unknowns or variables. The hotel rates example is a practical application of algebra where the total cost for rooms cannot be known until the quantity of each room type is considered. You may not always think of it as an algebra problem when calculating such costs, defending budgets, or planning expenses, but these are indeed applications of algebra. By identifying the variables (number of rooms and their rates), you can set up an equation or expression. You then apply the rules of algebra, including order of operations and grouping symbols, to solve the problem, just as one would to find out the exact cost for different groups of people staying at a hotel.