Problem 7 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$\frac{-14}{-7}$$

Step-by-Step Solution

Answer

The quotient is 2.

Step 1: Understanding the Problem

We need to find the quotient by dividing two negative numbers: \(-14\) divided by \(-7\).

Step 2: Identify the Signs

Note that both the numbers (\(-14\) and \(-7\)) are negative. When dividing two numbers, if both have the same sign, the result will be positive.

Step 3: Divide the Absolute Values

Dividing the absolute values of the numbers: \(|-14| = 14\) and \(|-7| = 7\). Calculate \(\frac{14}{7} = 2\).

Step 4: Apply Sign Rule

Since both numbers were negative, the quotient will be positive. Therefore, \(\frac{-14}{-7} = 2\).

Step 5: Write the Final Answer

The quotient of \(-14\) divided by \(-7\) is \(+2\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They can be identified by the minus sign ("-") in front of the number. These numbers play a crucial role in mathematics because they help account for phenomena such as debts, losses, or temperatures below zero. Understanding negative numbers is essential for solving many math problems, especially those involving operations like multiplication or division. When you multiply or divide negative numbers, the result can change dramatically depending on the signs involved. The rules governing these operations may seem tricky at first, but they become quite intuitive with practice.

Absolute Values

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number, as it represents only magnitude, not the direction. The absolute value of a number is denoted by vertical bars around the number. For example, the absolute value of

• -14 is written as \( |-14| = 14 \).

Similarly,

• \( |-7| = 7 \).

When dividing negative numbers, we often work with absolute values to simplify calculations before applying sign rules. This approach reduces the equation to simple positive number division and separately considers how the signs affect the result.

Division Rules

Dividing two numbers involves determining how many times the divisor fits into the dividend. The basic division rule states that

• the dividend is divided by the divisor.
• For instance, in \( \frac{-14}{-7} \), we treat it as \( \frac{14}{7} \) using absolute values.

This results in

• 2, since 7 goes into 14 exactly two times.

Understanding division rules allows us to perform division operations correctly and is crucial in simplifying expressions and solving equations where division is involved.

Sign Rules

Sign rules are fundamental in determining the result of operations like multiplication and division. The rules for signs are simple:

• If you divide two numbers with the same sign (both positive or both negative), the result is positive.
• Conversely, if they have different signs, the result is negative.

In our example, both

• \(-14\) and \(-7\) are negative,

so their quotient, calculated using absolute values as

• \( \frac{14}{7} = 2 \),

is positive. Remembering sign rules helps ensure that calculations are both accurate and consistent across different problems.