Problem 110 - Prealgebra and Introductory Algebra

Question

What is the product of \(0.06,-0.4,\) and \(-1.5 ?\)

Step-by-Step Solution

Answer

The product of \(0.06, -0.4\), and \(-1.5\) is \(0.036\).

Step 1: Identify the given numbers

The numbers given in the problem are \(0.06, -0.4\) and \(-1.5\). It's noted that two of these numbers are negative.

Step 2: Perform multiplication

Multiply the given numbers in any order since the operation of multiplication is commutative. \(0.06 * -0.4 * -1.5\).

Step 3: Calculate the result

Performing the multiplication gives the result \(0.06 * -0.4 * -1.5 = 0.036\)

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are often shown with a minus sign (-) in front of them. When we deal with negative numbers, it is important to understand how they behave in mathematical operations because they affect the outcome in unique ways.
For instance, when multiplying negative numbers:

• Multiplying a positive number by a negative number results in a negative product.
• Multiplying two negative numbers traditionally turns the product positive.

These rules come from the definition and properties of negative numbers. In the original exercise, the numbers -0.4 and -1.5 are both negative, and when they are multiplied together, they result in a positive number, despite each individually being negative.

The Commutative Property of Multiplication

The commutative property is a fundamental property of multiplication and states that the order of numbers in a multiplication operation does not affect the product. This property is tremendously useful in calculations as it allows us to rearrange numbers for easier computation.
For example, if you need to calculate the product of 0.06, -0.4, and -1.5, you can choose any order that is most convenient, such as:-0.4 multiplied by -1.5, and then multiplied by 0.06. This flexibility is crucial, especially when dealing with multiple numbers or complex calculations, allowing you to simplify expressions by rearranging factors.

Exploring Mathematical Operations

Mathematical operations include actions such as addition, subtraction, multiplication, and division. Each operation follows specific rules and properties, which determine the outcomes of calculations.
Multiplication, for example, involves combining quantities, and it can include factors of positive and negative numbers, as well as decimals. Understanding these operations and properties is key in solving mathematical problems efficiently and correctly.

• Addition combines numbers to get a sum.
• Subtraction finds the difference between numbers.
• Multiplication aggregates based on repeating additions.

In the given problem, multiplication is the operation used to find the product of all the numbers listed, including handling negative values wisely to derive the correct answer.

Decimal Multiplication Basics

Multiplying decimals is very similar to multiplying whole numbers. However, extra care is required with the placement of the decimal point in the product. To multiply decimals, we can follow these steps:

• First, ignore the decimals and multiply the numbers as if they were whole numbers.
• After obtaining the product, count the total number of decimal places in the factors. This number will determine how many places to move the decimal point in the final product.

Applying these steps to the original problem, consider 0.06, -0.4, and -1.5. After tackling signs due to negatives, treat them as 6, 4, and 15. Multiply them, resulting in 360. Since there are four decimal places in total across all numbers (two in 0.06 and one each in both -0.4 and -1.5), adjust the result to 0.036.