Problem 28 - Fundamentals of Mathematics

Question

Find the value of each of the following. Use a calculator to check each result. $$ \frac{-20}{10} $$

Step-by-Step Solution

Answer

The value is -2.

Step 1: Identify the Division

In the expression \( \frac{-20}{10} \), we need to divide -20 by 10. The negative sign indicates that the result should also be negative.

Step 2: Simplify the Division

Divide the absolute values of the numbers first. \( 20 \div 10 = 2 \). Since the original numerator is negative, the final result will be negative as well.

Step 3: Apply the Sign

Attach the negative sign to the result from the previous step. Thus, the division of \( -20 \) by \( 10 \) is \( -2 \).

Key Concepts

Division of Integers

When we talk about dividing integers, we are essentially breaking down one integer into a specified number of smaller parts, as dictated by another integer. Take the expression \( \frac{-20}{10} \) for example. Here, you are dividing \(-20\) by \(10\). This means you want to find out how many times \(10\) fits into \(-20\).

To start, divide the absolute value of both numbers, which means ignoring the negative sign for a moment. \( 20 \div 10 \) gives you \(2\). Remember, division involves both determining the magnitude and the sign of the solution.

Since one of the numbers is negative, the answer will naturally carry a negative sign too. This means that the expression \( \frac{-20}{10} \) simplifies to \(-2\). It is crucial to remember this step – the negative sign indicates that you are moving in the opposite direction on the number line, hence the negative result.

• When both numbers have the same sign, the result is positive.
• When the numbers have different signs, the result is negative.

Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign before them. For example, in the fraction \( \frac{-20}{10} \), \(-20\) is a negative number.

Negative numbers can be somewhat tricky, but they follow a consistent set of rules that make them manageable.

• When you multiply or divide an odd number of negative numbers, the result will be negative.
• When you multiply or divide an even number of negative numbers, the result will be positive.

Understanding and remembering these rules makes dealing with negative numbers much more straightforward. In our example, since \(-20\) is being divided by \(10\), their different signs lead to a negative result as per the rules listed above. Negative numbers are integral to arithmetic and appear often in real-world scenarios, like debts and temperatures below zero.

Simplifying Fractions

Simplifying fractions is about reducing the fraction to its smallest possible equivalent by dividing its numerator and the denominator by their greatest common divisor. In the problem \( \frac{-20}{10} \), simplifying means recognizing the common factors of the numerator and the denominator.

Here's how you do it:

1. Identify the greatest common divisor of \( 20 \) and \( 10 \), which is \( 10 \).
2. Divide both the top and bottom of the fraction by \( 10 \).

When you do so, \( \frac{-20}{10} \) simplifies to \(-2\) because once you divide both values, the fraction reduces directly to a whole number.

It's important to note that simplification does not alter the value of the fraction; it just makes the fraction easier to work with. By simplifying, you still maintain the original value despite changing its appearance.

• Always simplify when possible, as it can make calculations much easier.
• To simplify effectively, always look for the largest factor common to both the numerator and the denominator.