Problem 85 - Prealgebra

Question

Use the rule for order of operations to combine the following. \(4 \cdot 5-3 \cdot 2\)

Step-by-Step Solution

Answer

The result is 14.

Step 1: Apply the Multiplication

First, we perform the multiplication operations according to the order of operations (PEMDAS/BODMAS), which prioritizes Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).Calculate the first multiplication: \(4 \cdot 5 = 20\)

Step 2: Perform the Second Multiplication

Next, calculate the second multiplication:\(3 \cdot 2 = 6\)

Step 3: Perform Subtraction

Using the results from the earlier steps, perform the subtraction as per the order of operations.Subtract: \(20 - 6 = 14\)

Key Concepts

Multiplication

In mathematics, multiplication is one of the basic operations that involves combining equal groups to find a total. It is an arithmetic operation where a number is added to itself a certain number of times. For example, in the expression "\(4 \cdot 5\)", you are adding the number 4, five times (\(4 + 4 + 4 + 4 + 4\)) to get 20. This is often seen as a faster way of adding the same number repeatedly.

Understanding multiplication as repeated addition can be particularly helpful when you're just learning the concept, as it provides a clear and tangible picture of what multiplication accomplishes. Here's a quick tip:

• The first number in the multiplication (4) is called the 'multiplier', and the number you multiply it by (5) is the 'multiplicand'.
• The result of this operation is called the 'product' (20).

Mastering multiplication is crucial as it is a fundamental operation used in more complex mathematical calculations.

Subtraction

Subtraction is another fundamental arithmetic operation, which involves taking away one number from another. In the context of this exercise, subtraction helps us find the difference between two products obtained from the multiplication operations.

For the expression "\(20 - 6\)", you are essentially finding out how much more 20 is than 6, or what remains when you remove 6 from 20. This is what subtraction always seeks to determine: the disparity or leftover amount. A few essential points to keep in mind:

• The first number (20) is the 'minuend'. It represents the amount you start with.
• The number you subtract (6) is the 'subtrahend'. It’s what you are taking away.
• The answer to the subtraction operation is called the 'difference' (14).

Subtraction is not only used in standalone situations but also functions in harmony with other operations like multiplication and addition, as you see in the order of operations.

PEMDAS

PEMDAS is a memorable acronym that students use to remember the order of operations in mathematics. It stands for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). This order is crucial because it dictates the sequence in which operations should be carried out to ensure the accurate evaluation of mathematical expressions.

In the expression \(4 \cdot 5 - 3 \cdot 2\), PEMDAS helped determine the sequence:

• First, perform the Multiplications: \(4 \cdot 5\) and \(3 \cdot 2\).
• Next, resolve the Subtraction: \(20 - 6\).

Without following PEMDAS, one could mistakenly perform the subtraction before the multiplication, leading to incorrect results. Using PEMDAS ensures clarity and consistency, critical for solving both simple and complex expressions in mathematics.