Problem 52 - Algebra 1: Concepts and Skills

Question

Simplify. $$ (-15)\left(-\frac{5}{6}\right) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is 12.5.

Step 1: Identify the numbers

Two numbers to multiply are (-15) and \(-\frac{5}{6}\). Both numbers are negative.

Step 2: Apply the rule of signs

The product of two negative numbers is a positive number. So, the result of multiplying these two numbers will be positive.

Step 3: Multiply the whole number by the fraction

Multiply the whole number (-15) by the numerator of the fraction (-5). That gives you 75.

Step 4: Divide by the denominator

Lastly, we need to divide the result obtained from step 3 by the denominator of the fraction (6). So, \( \frac{75}{6} \) equals 12.5

Key Concepts

Rules of Signs

Understanding the rules of signs is crucial when dealing with multiplication of negative numbers. According to these rules, multiplying two negative numbers together will always yield a positive result. Conversely, the product of a positive number and a negative number will always be negative. When multiplying multiple numbers together, count the number of negative signs: if there's an even number of them, the product is positive; if there's an odd number, the product is negative.

This rule is not only foundational in arithmetic, but it's also fundamental in more advanced areas of math such as algebra, where you deal with variables. Regardless of whether you're multiplying simple integers or complex polynomials, the rules of signs remain consistent and reliable.

Multiplying Fractions

Step-by-Step Multiplication

To multiply fractions, you don't need a common denominator as you do with addition or subtraction. Instead, you multiply the numerators together to get a new numerator, and the denominators together to get a new denominator. Here's the simplified process:

• Multiply the numerators.
• Multiply the denominators.
• Simplify the result, if possible.

When dealing with mixed numbers, you first convert them to improper fractions before following these steps.

Dealing with Whole Numbers

If you're multiplying a fraction by a whole number, consider the whole number as a fraction over 1 and then follow the same procedure. The multiplication process simplifies arithmetic operations and helps in breaking down more complex equations.

Simplifying Expressions

Simplifying expressions refers to the process of making an algebraic, numerical, or arithmetic expression as simple as possible. The aim is to reduce the complexity while keeping the expression equivalent to the original. Simplification might involve

1. Combining like terms.
2. Using the distributive property.
3. Reducing fractions to their lowest terms.
4. Applying exponent rules.

Simplification is not merely for making an expression look neater; it often reveals the core components of an expression, making it easier to work with, especially when you're solving equations or inequalities. The ability to simplify an expression is a key skill that can dispel confusion and help students better understand and solve mathematical problems.

Arithmetic Operations

Arithmetic operations are the cornerstone of math, comprising addition, subtraction, multiplication, and division. These operations are performed following a specific order known as the order of operations, often memorized with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

When performing arithmetic operations, especially multiplication and division, it's essential to be mindful of the rules of signs. Different operations have different properties that affect how numbers interact with each other, such as the commutative property for addition and multiplication, or the associative property for grouping numbers.

Understanding how to adeptly navigate through these operations is more than a mathematical requirement; it's a basic tool that empowers problem-solving skills across academic and real-world applications.