Problem 38 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{-20}{10} $$

Step-by-Step Solution

Answer

Answer: -2

Step 1: Identify the numbers involved in the fraction

Here, we have a fraction with -20 as the numerator and 10 as the denominator. So our expression looks like this: $$ \frac{-20}{10} $$

Step 2: Rules for dividing integers

When dividing integers, we must consider the signs of both numbers. In this case, the numerator is negative and the denominator is positive. When dividing a negative integer by a positive integer, the result will be a negative value.

Step 3: Perform the division

Now, we simply need to divide the absolute values of the two numbers and assign the correct sign to the result. So we have: $$ \frac{-20}{10} = -\frac{20}{10} $$ Dividing 20 by 10, we get 2. Remembering to keep the negative sign from Step 2, our final result is: $$ -\frac{20}{10} = -2 $$ So the value of the given expression is -2.

Key Concepts

Understanding Negative Numbers in Division

Negative numbers can seem tricky at first, but they follow simple rules that make them easier to handle. In mathematics, a negative number is a number that is less than zero. It is represented with a minus sign (-) right before the number.- **Sign Matters:** When performing operations with negative numbers, especially division, the sign plays a crucial role. - **Result of Negatives in Division:** When dividing a negative number by a positive number, the result will always be negative. Similarly, if a positive number is divided by a negative number, the result will be negative too. On the other hand, a negative number divided by another negative gives a positive result.For example, in the expression \( \frac{-20}{10} \), the negative sign from the numerator ensures that the final result will be negative. When dividing, you simply ensure the negative sign carries through to the result, reaffirming the rule: **Negative divided by Positive is Negative.**
Just remember, when dealing with negative numbers, a clear understanding of the signs will guide you to the correct answer.

Fractions in Division

Fractions represent a part of a whole and are composed of a numerator (top part) and a denominator (bottom part). When we see a fraction in division like \( \frac{-20}{10} \), it means dividing -20 by 10.- **Fractions as Division:** At its core, a fraction is essentially a division expression. The numerator is the number to be divided, and the denominator is the number by which you divide it.- **Simplifying Fractions:** In any division involving fractions, the ultimate goal is to simplify it if possible. This means reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).In our original example, the greatest common divisor of 20 and 10 is 10. By simplifying \( \frac{-20}{10} \) using the GCD, we divide both by 10, resulting in \( -2 \). This is how fractions help in breaking down division into simpler terms.

Exploring Division Rules

When working with division, understanding the basic rules is essential to solving problems correctly:- **Basic Division Rule:** Division is the process of finding how many times one number is contained within another. It’s a foundational operation in arithmetic used to break down numbers into smaller, more manageable parts.- **Sign Rules in Division:** As previously mentioned, the sign rules in division are crucial when dealing with integers:

• Positive ÷ Positive = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative
• Negative ÷ Negative = Positive

- **Using Absolute Values:** A helpful strategy is to temporarily focus on the absolute values (ignore the signs) of numbers, perform the division, and then apply the appropriate sign based on the division sign rule.In the exercise \( \frac{-20}{10} \), after ignoring the signs, we first divided 20 by 10 to get 2, then applied the division sign rule, giving us -2. Memorizing these simple rules can make division straightforward and less intimidating, regardless of the numbers’ signs.