Problem 40 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The result of the operation is -9.

Step 1: Understand the Division Problem

The problem requires us to divide \(-6.3\) by \(0.7\). This means we need to find out how many times \(0.7\) fits into \(-6.3\).

Step 2: Simplify the Division by Removing Decimals

To simplify the division \(\frac{-6.3}{0.7}\), we can remove the decimals by multiplying both the numerator and the denominator by 10. This changes the problem to \(\frac{-63}{7}\).

Step 3: Perform the Division

Now divide \(-63\) by \(7\). Since \(7 \times (-9) = -63\), we can see that \(-63\) divided by \(7\) equals \(-9\).

Step 4: Confirm the Result

To confirm the result, multiply \(-9\) by \(0.7\) to verify it equals \(-6.3\). Since \(-9 \times 0.7 = -6.3\), the division result is confirmed correct.

Key Concepts

Division of Real Numbers

When we're dealing with the division of real numbers, our goal is to determine how many times one number, called the divisor, fits into another number, the dividend. In our example, we are dividing \(-6.3\) by \(0.7\). This involves understanding the basic principles of division.

• Divisor: The number we are dividing by, in this case, \(0.7\).
• Dividend: The number to be divided, which is \(-6.3\).
• Quotient: The result of the division.

When dividing with decimals, it often helps to simplify the equation. Changing the dividend and divisor into whole numbers can make division easier to understand and compute.
In our specific case, multiplying both numbers by 10 shifts the decimal place, transforming our equation from \(-6.3 \div 0.7\) into \(-63 \div 7\).

Simplifying Fractions

Simplifying fractions is about making them as simple as possible by reducing them to their lowest terms. In mathematics, working with simpler numbers can decrease errors and improve clarity.
Here are simple steps to simplify a fraction:

• Find the Greatest Common Factor (GCF): The highest number that divides both the numerator and the denominator without leaving a remainder.
• Divide: Reduce the fraction by dividing both the numerator and the denominator by their GCF.

In our example, when the fraction \(\frac{-6.3}{0.7}\) is transformed into \(\frac{-63}{7}\), it demonstrates a simplified form because the GCF of 63 and 7 is 7. Dividing both by 7 gives \(-9\), simplifying the operation and solution.

Negative Numbers Operations

Operations with negative numbers can sometimes be confusing, so let's break them down. When performing operations like division, it's essential to follow the rules for negative numbers:

• Multiplying and Dividing: When multiplying or dividing two real numbers:
  • If both numbers have the same sign (e.g., both are negative, or both are positive), the result is positive.
  • If the numbers have different signs, the result is negative.
• Sign Awareness: Always pay attention to the sign before a number, as it affects the division outcome.

In our operation, \(-63\) divided by \(7\) equals \(-9\) because one number is negative and the other is positive. Understanding these rules can help avoid mistakes and ensure accurate calculations in operations involving negative numbers.