Problem 53 - Prealgebra

Question

Find the value of each expression. $$5(16-5 \times 3)$$

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Understand the Order of Operations

Before solving the expression, recall the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This will guide how you approach simplifying the expression.

Step 2: Solve Inside the Parentheses

In the expression, identify the part within the parentheses: \(16 - 5 \times 3\). According to the order of operations, solve the multiplication inside the parentheses first. Calculate \(5 \times 3 = 15\).

Step 3: Complete the Operations in the Parentheses

Replace the multiplication with its result in the parentheses, so the expression becomes \(16 - 15\). Now perform the subtraction: \(16 - 15 = 1\).

Step 4: Multiply by the Number Outside the Parentheses

Now return to the main expression which was \(5(16-5 \times 3)\). After solving inside the parentheses, it simplifies to \(5 \times 1\). Now, perform the multiplication: \(5 \times 1 = 5\).

Key Concepts

PEMDAS

A key tool in solving math problems is the order of operations. The order of operations tells you the correct sequence to solve different parts of a math equation. It's often remembered through the acronym PEMDAS. This stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Whenever you address an equation, use PEMDAS to decide which operations to perform first. By following this sequence, you'll avoid mistakes and find the correct answer. The main idea is to handle any operations inside parentheses first, then tackle any exponents, and so on. This helps in maintaining the logical and mathematical integrity of your work. For example, in the expression given, starting with the parentheses ensures that each operation builds correctly on the last.

Multiplication

Multiplication is one of the core operations you need to handle when solving math expressions. In PEMDAS, multiplication is performed after parentheses and exponents, but before addition and subtraction. It's also crucial to process multiplication and division from left to right as they appear in the expression.
In our exercise, once we solve inside the parentheses, we encounter the operation: multiplication. Correctly processing this is essential for progressing through the equation.
The expression inside the parentheses was initially:

• 5 times 3 equals 15, which is calculated as:

\(5 \times 3 = 15\).
After this, multiplication appears again outside the parentheses when you find:

• 5 times 1 which results in the final multiplication operation in the problem:

\(5 \times 1 = 5\).
Remember, properly executing multiplication in the correct sequence ensures the accuracy of your work.

Expressions

In mathematics, expressions are combinations of numbers, variables, and operations that define a particular calculation. When solving such expressions, deciphering the order and types of operations involved is crucial.
Expressions can be simple or complex, depending on how many components they include. The expression in our exercise,

• \(5(16 - 5 \times 3)\),

exemplifies a compound expression due to multiple operations. First, you'll notice parentheses, dictating that the operations enclosed must be handled first.
Expressions come in various forms, like algebraic expressions (with variables) or numerical expressions (without variables, like ours). Solving them requires following the right sequence of operations, making sure each step follows logically and consistently from the previous one. This maintains the mathematical structure and integrity of the problem, leading you to the correct answer.

Parentheses

Parentheses are a crucial component of mathematical expressions. They help in grouping parts of an expression that should be treated as a single entity. When parentheses appear in an expression, they signal that the operations inside should be completed before anything else.
In our provided exercise, the expression starts with parentheses:

• \(5(16-5 \times 3)\)

The parentheses contain the expression \(16-5 \times 3\), which must be resolved first according to PEMDAS. Calculating the multiplication within the parentheses first provides clarity and assures the rest of the operations can proceed correctly.
By first solving inside the parentheses, \(5 \times 3 = 15\), and then \(16 - 15 = 1\), you ensure the expression is simplified step by step.
This structured approach ensures that when you finally perform operations outside the parentheses, like the multiplication \(5 \times 1\), the accurate simplification leads directly to the correct answer. Parentheses are vital in guiding the orderly and logical resolution of complex expressions.