Problem 7 - Elementary Algebra

Question

Find the following quotients. $$ \frac{-24}{-6} $$

Step-by-Step Solution

Answer

Answer: The quotient of the fraction $\frac{-24}{-6}$ is 4.

Step 1: Identify the sign of the result

Since both the numerator and the denominator are negative, the result will be positive.

Step 2: Divide the absolute values of the numerator and the denominator

Now, we need to divide the absolute values of the given numbers: $$ \frac{| -24 |}{| -6 |} = \frac{24}{6} $$

Step 3: Calculate the division

We can now perform the division for the given fraction: $$ \frac{24}{6} = 4 $$

Step 4: State the final answer

As we determined in Step 1, the result will be positive. Therefore, the final answer is: $$ \frac{-24}{-6} = 4 $$

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is essential for mastering arithmetic operations. In the context of division, it's important to know how these signs affect the outcome. A positive number is a number greater than zero and is often represented without a sign or with a plus sign (+) before it. Meanwhile, a negative number is less than zero, typically indicated by a minus sign (−) in front. When dividing integers with different signs, the result is negative; conversely, if they share the same sign, the result is positive.

For instance, in the exercise \( \frac{-24}{-6} \), both numbers are negative. When dividing numbers with the same sign, the negatives cancel each other out. Hence, the quotient of two negative numbers is a positive number. Had it been \( \frac{-24}{6} \), the quotient would be negative, since the numbers have opposite signs.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Represented by two vertical lines (| |), it transforms negative numbers to positive. Thus, \( | -24 | \) becomes \( 24 \) because \( -24 \) is 24 units away from zero. Absolute values are useful in arithmetic operations, particularly in division, as they simplify the problem to basic number division. In our example, we simplify to \( \frac{| -24 |}{| -6 |} = \frac{24}{6} \) before calculating the division, which further aids in understanding the nature of the numbers involved and ensures accurate results after considering the initial signs.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These fundamental concepts are the building blocks for more complex mathematics. Division, our focus here, involves splitting a number (the dividend) into equal parts, as specified by another number (the divisor). The outcome is the quotient. When performing division, it's crucial to discern whether the numbers are positive or negative, as this affects the sign of the quotient. For simplification, after determining the sign, we deal with the absolute values to execute the division, which is a straightforward operation once the concept of absolute value is clear.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This makes understanding and working with fractions easier. To simplify a fraction, both the numerator and denominator should be divided by their greatest common divisor (GCD). In our exercise, we simplified the fraction \( \frac{24}{6} \) to get \( 4 \) because 24 and 6 are both divisible by 6. By simplifying fractions, we not only achieve the most reduced form but also make it easier to interpret the result in contexts like comparing quantities or converting fractions to decimals.