Problem 86 - Prealgebra

Question

Use the rule for order of operations to combine the following. \(9 \cdot 7+6 \cdot 5\)

Step-by-Step Solution

Answer

The result is 93.

Step 1: Apply the Order of Operations

To solve the expression \(9 \cdot 7 + 6 \cdot 5\), we must first apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this expression, multiplication takes precedence over addition.

Step 2: Perform Multiplication

First, compute the products: \(9 \cdot 7 = 63\) and \(6 \cdot 5 = 30\). Now the expression becomes \(63 + 30\).

Step 3: Perform Addition

Next, perform the addition: \(63 + 30 = 93\). This is the final step in solving the expression.

Key Concepts

PEMDAS

The order of operations is vital in mathematics to ensure everyone solves problems consistently and correctly. You've probably heard of the trusty acronym PEMDAS to remember this rule. PEMDAS stands for:

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

In any mathematical expression, these rules dictate which operations to perform first. Without following PEMDAS, you might end up with totally wrong results! When looking at an expression like our original example, you would perform any calculations inside parentheses first. Then, if there are any exponents, you'd handle those next. The most common point of confusion arises because Multiplication and Division or Addition and Subtraction are done from left to right, not necessarily multiplication before division or addition before subtraction. Following PEMDAS ensures that everyone speaks the same mathematical language, essential for correct problem solving.

Multiplication

Multiplication is an operation where you calculate the total of one number repeated a certain number of times. In the context of the order of operations, multiplication has a higher priority than addition or subtraction, so you tackle it first. Looking at the expression given, there's multiplication to be done for both sets of numbers.

1. Multiply 9 by 7 to get 63.
2. Multiply 6 by 5 to get 30.

Once you solve these multiplications, the expression becomes much simpler. Multiplying first before adding or subtracting helps keep math neat and easy to understand, as it breaks down complex expressions into manageable parts.

Addition

Addition is often introduced as the first arithmetic operation learned in math, it's the process of finding the total or sum by combining two numbers. After multiplication is confirmed in the order of operations, addition then completes the calculations.In our simplified expression of \(63 + 30\) only addition remains to be performed. Simply add together 63 and 30, which totals to 93.This step can seem simple, but it's crucial to get right. Knowing when to apply addition, after other operations are completed, is key in solving arithmetic problems correctly. Making addition the finishing touch, assures the calculations hold up under PEMDAS rules.

Prealgebra

Prealgebra lays the foundation for all of algebra and beyond, focusing on operations and properties of numbers. It's where mathematical concepts such as PEMDAS become part of a student's toolkit. This is the stage when learners understand how numbers work together in expressions. Prealgebra helps identify which parts of a problem to solve first, much like our simple expression involving multiplication and addition. Prealgebra isn't just about arithmetic; it's about understanding how arithmetic fits into more complex math concepts. For instance, learning the role of multiplication and addition under PEMDAS helps students transition smoothly into algebra with confidence in their foundational skills. Applying these concepts through prealgebra, students gain the confidence and ability to tackle more advanced problems with an organized approach, reducing errors and improving their overall math fluency.