Problem 43 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ \frac{-1.2}{-6} $$

Step-by-Step Solution

Answer

The result is 0.2.

Step 1: Identify the operation

We are given the expression \( \frac{-1.2}{-6} \). This represents a division operation between two negative real numbers, \(-1.2\) and \(-6\).

Step 2: Apply the rule for division of negative numbers

The rule for dividing two negative numbers is that the negatives cancel out, resulting in a positive number. Therefore, \( \frac{-1.2}{-6} \) becomes \( \frac{1.2}{6} \).

Step 3: Perform the division

Now, take the absolute values and perform the division: divide 1.2 by 6. To do this, convert the decimal to a fraction or perform direct division. Calculating, \( 1.2 \div 6 = 0.2 \).

Key Concepts

Division of Negative Numbers

Dividing negative numbers may seem tricky, but it's straightforward once you know the rule. When you divide two negative numbers, the result is always positive. This is because a negative sign indicates a reversal of direction or subtraction. When two of these negative signs interact, they neutralize each other, leading to a positive result.
This applies universally, whether the numbers are whole numbers, fractions, or decimals. It's essential to always remember this rule when dividing any two negative numbers to avoid mistakes.

• If both the dividend and the divisor are negative, the quotient is positive.
• Example: For \(-a \/ -b = a \/ b\).

Remember, this property stems from the broader rule in mathematics that negating a negative reverses it back to positive. So, in our example, when you divide \(-1.2\) by \(-6\), the result is positive 0.2.

Operations with Real Numbers

Real numbers include a vast set of numbers such as integers, fractions, and decimals. They form the building blocks of everyday arithmetic and include both positive and negative numbers as well as zero.
When performing operations like addition, subtraction, multiplication, and division with these numbers, the same rules apply as with integers, but with more care. For instance:

• Addition/Subtraction: Add real number values as you usually would, keeping a note of their respective signs.
• Multiplication: Real numbers multiply as expected, with the product's sign depending on the sign of the numbers multiplied.
• Division: When dividing, pay attention to the signs as well as the decimal places.

Understanding that real numbers cover a broad spectrum should encourage precision and care when conducting any arithmetic operations with them.

Decimal Division

Performing division with decimals involves some additional steps compared to whole numbers. It requires careful placement and attention to decimal points to ensure accuracy. Here's a simple guide:

• Step One: If dividing a decimal by a whole number, you can perform division as usual and place the decimal point directly above in your answer model.
• Step Two: When dividing a decimal by a decimal, first move the decimal point in the divisor to the right until it becomes a whole number. Do the same shift in the dividend simultaneously.
• Step Three: Divide as if working with whole numbers and place the decimal point in the result directly above its position in the adjusted dividend.

In our given problem, we divided \(1.2 \/ 6\), resulting in 0.2, illustrating how straightforward managing decimals in division can be once you follow the steps closely.