Problem 49 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{-12}{-4}\)

Step-by-Step Solution

Answer

The result is 3.

Step 1: Identify the Operation

The exercise requires us to perform division. The expression is a fraction, where -12 is the numerator and -4 is the denominator.

Step 2: Simplify the Negative Signs

When both the numerator and the denominator are negative, they can be simplified by removing the negative signs. This simplification is due to the rule that the division of two negative numbers results in a positive result. Therefore, the fraction becomes \(\frac{12}{4}\).

Step 3: Perform the Division

Now, perform the division of the simplified fraction. Divide 12 by 4, which equals 3.

Key Concepts

Division

Division is one of the four basic arithmetic operations. It is the process of determining how many times one number, the divisor, is contained within another number, the dividend. The result of division is called the quotient. In mathematical expressions, division is often represented with a forward slash "/" or as a fraction line.

When working with fractions, division can be simplified by thinking of it as distributing the numerator across the denominator. Here, in the expression \(\frac{-12}{-4}\), you divide -12 by -4. This gives us the number of times -4 fits into -12. Understanding division is essential because it lays the foundation for more complex mathematical concepts.

• Dividends are the numbers being divided.
• Divisors are the numbers you divide by.
• Quotients are the results of division.

The key to mastering division is practice, which helps in recognizing number patterns and relationships.

Negative Numbers

Negative numbers can be a bit confusing, but they're very important in arithmetic. These are numbers less than zero, represented with a minus sign "-". In real life, they can represent debts or temperatures below zero.

When dividing negative numbers, it's crucial to remember a basic rule: dividing two numbers with the same sign (both positive or both negative) results in a positive number. Hence, dividing two negative numbers, such as -12 and -4, results in a positive outcome.

• A negative divided by a positive results in a negative.
• A negative divided by a negative results in a positive.
• Similarly, a positive divided by a negative is negative.

Understanding how negative numbers interact helps prevent mistakes in computations, especially in tasks like simplifying fractions.

Simplifying Fractions

Simplifying fractions is an important skill because it makes working with numbers more manageable. This process involves reducing the fraction to its simplest form where the numerator and denominator have no common factors other than 1.

In our example, \(\frac{-12}{-4}\), you first simplify the negatives as discussed previously, turning it into \(\frac{12}{4}\).

Next, simplify by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here, both 12 and 4 can be divided by 4, giving us a simplified fraction: \(\frac{3}{1}\), which is simply 3.

• Identify any common factors between the numerator and the denominator.
• Divide both by their GCD for a simplified fraction.
• A fraction in simplest form makes it easier to understand and use.

Simplifying fractions not only makes math easier but also helps in comprehending more complex problems.