Problem 45 - Prealgebra

Question

Copy each sentence. Then insert parentheses to make each sentence true. $$5+2 \cdot 9-3=42$$

Step-by-Step Solution

Answer

Place parentheses as \((5 + 2) \cdot (9 - 3)\) to make it true.

Step 1: Identify the Order of Operations

To properly solve the expression, identify the standard order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This order can be remembered by the acronym PEMDAS.

Step 2: Explore Possible Parentheses Placements

Analyze different ways to place parentheses in the expression to change the order of operations. Consider the placement of parentheses around different operations or numbers to attempt to make the sentence true: \(5 + (2 \cdot 9) - 3 = 42\), \((5+2) \cdot (9-3) = 42\), \((5+2 \cdot 9) - 3 = 42\), etc.

Step 3: Test Parentheses Around Multiplication First

Test the expression where multiplication is done first, as is original without parentheses: \(5 + (2 \cdot 9) - 3\). Simplifying gives \(5 + 18 - 3\), which equals 20, not 42.

Step 4: Test Parentheses Around Entire Expression

Try adding parentheses around the entire multiplication and subtraction: \((5 + 2) \cdot (9 - 3)\). Simplify the expression inside the parentheses: \((7) \cdot (6)\), which equals 42. This expression is true.

Key Concepts

PEMDAS

Understanding the order of operations is essential when solving mathematical expressions. The acronym PEMDAS helps us remember which operations to perform first. Each letter in PEMDAS stands for a specific operation:

• P - Parentheses: Solve expressions inside parentheses first.
• E - Exponents: Next, calculate any exponents.
• M and D - Multiplication and Division: These are performed from left to right.
• A and S - Addition and Subtraction: Lastly, execute these operations from left to right.

By using PEMDAS, you can approach any arithmetic expression systematically, ensuring that you achieve the correct result. It’s like following a recipe; each step needs to be completed in a specific order to avoid errors.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, and operations. It represents a calculation or a value. For instance, in the example problem, the expression given is:\[5 + 2 \cdot 9 - 3 = 42\]This is an expression with numbers and operations that need to be evaluated to determine if it equals 42. However, without proper guidance in operation sequence, calculating the correct result might not be straightforward. Expressions exist to solve problems or express mathematical ideas. To simplify or solve expressions:

• Identify all components like numbers and operations.
• Use PEMDAS to determine the order of operations.
• Break down complex parts into simpler segments using parentheses to guide the process.

Parentheses Use

Parentheses play a crucial role in altering the sequence or grouping within mathematical expressions. When you place parentheses in an expression, you define which operations should be performed first. By doing so, you can entirely change the expression's outcome. In the exercise provided, different placements of parentheses were explored to achieve the correct result, 42:

• Initially, evaluating without parentheses: \[ 5 + 2 \cdot 9 - 3 = 20 \] didn’t give the desired result.
• By adjusting the parentheses to change the evaluation order: \[ (5 + 2) \cdot (9 - 3) = 42 \] the expression calculates correctly.

Thus, by strategically placing parentheses, we control the calculations and ensure they yield the intended result. Parentheses act like cups, holding certain operations together, so they are prioritized in calculations.