Problem 20 - Algebra 1: Concepts and Skills

Question

Find the quotient. \begin{equation} -10 \div(-5) \end{equation}

Step-by-Step Solution

Answer

The quotient of \(-10 \div -5\) is 2.

Step 1: Identify the Numbers and Operation

The given expression is \(-10 \div(-5)\). It is a simple division operation where -10 is being divided by -5.

Step 2: Apply the Division Operation

When a negative number is divided by another negative number, the result is a positive number. So, \(-10 \div -5 = 2\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are often used to represent loss, decrease, or values below a standard reference point in mathematics. When working with negative numbers, especially in division, it's important to remember how they interact with each other.

• Dividing two negative numbers results in a positive number. This is because a negative divided by a negative flips the negative sign, resulting in a positive quotient.
• Negative numbers have special rules in arithmetic operations. For instance, a negative number divided by a positive number results in a negative quotient.

Understanding these interactions is crucial for solving equations that involve negative values accurately. This means if you divide -10 (a negative ten) by -5 (a negative five), according to these rules, the outcome is positive.

Quotient

A quotient is the result of a division operation. In the expression \(-10 \div -5\), the quotient is 2. The division operation can be thought of as asking, "How many times does \(-5\) fit into \(-10\)?" The answer is 2 times.

• Quotients can be whole numbers or fractions/decimals, depending on whether the division is exact or not.
• When dividing negative numbers, always consider the rules for negative sign division to determine the sign of the quotient.

This problem results in a whole number because \(-5\) goes exactly into \(-10\) twice with no remainder.

Arithmetic Operations

Arithmetic operations are the basic functions of mathematics used for calculations. These include addition, subtraction, multiplication, and division. Each operation has its own rules and properties, especially when negative numbers are involved.

• **Addition**: Adding two negative numbers results in a more negative number.
• **Subtraction**: Subtracting a negative number is the same as adding its positive counterpart.
• **Multiplication**: Multiplying two negative numbers results in a positive product.
• **Division**: Dividing two negative numbers gives a positive quotient, as seen in \(-10 \div -5 = 2\).

Understanding these operations allows you to solve mathematical problems that involve various combinations of positive and negative numbers. They are foundational to more complex math concepts and everyday computations.