Problem 44 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ \frac{-6.3}{0.7} $$

Step-by-Step Solution

Answer

-9

Step 1: Identify the operation

The exercise asks us to divide -6.3 by 0.7.

Step 2: Consider the signs

Since the operation involves a division of two numbers, consider their signs: -6.3 is negative, and 0.7 is positive. The result of dividing numbers with different signs is negative.

Step 3: Simplify the numbers

To simplify the operation, think about the division as an equivalent whole number operation: -6.3 / 0.7 is equivalent to -63 / 7 after scaling both numbers by 10.

Step 4: Perform the division

Divide 63 by 7 which equals 9.

Step 5: Apply the sign

Since we determined that the result should be negative from step 2, the final result is -9.

Key Concepts

Division of Real Numbers

Division is the process of determining how many times one number is contained within another. When dealing with real numbers, which include both rational and irrational numbers, the division process remains the same as with integers.
Let's work through an example: the division of \(rac{-6.3}{0.7}\).

• First, we understand that we are dividing -6.3 by 0.7.
• The division can be interpreted as asking how many times 0.7 fits into -6.3.

To make calculations straightforward, it helps to simplify by scaling up. We can convert each number to whole numbers by multiplying both -6.3 and 0.7 by 10, giving us an equivalent operation: \(-63 / 7\). Division, thus, becomes easier without altering the result.

Negative and Positive Numbers

Understanding positive and negative numbers is crucial for operating manuevers like division. Here, it's essential to know how signs influence the outcome of your operations.

• A positive number lies above zero on the number line, whereas a negative is below zero.
• In division, like signs (positive/positive or negative/negative) result in a positive number.
• Opposite signs (positive/negative or negative/positive) yield a negative result.

In our example, -6.3 is a negative number, while 0.7 is positive.
The division has different signs, thus the result will be negative.

Simplifying Fractions

Simplifying fractions makes division less intimidating and more manageable. To simplify, aim to convert the numbers into easier forms without changing their value.

• In the exercise, -6.3 can be viewed as a fraction over another number, 0.7.
• Makes your job easier by scaling these numbers to remove decimals, turning -6.3 into -63 and 0.7 into 7, because both are multiplied by 10.
• Now, a simpler fraction \(-63/7\) can be divided directly to find the result.

Handling fractions effectively in division allows you to focus on simple arithmetic, ensuring the accuracy of your answer without decimals getting in the way.