Problem 49 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ -\frac{2}{3}-\frac{7}{9} $$

Step-by-Step Solution

Answer

The result is \(-\frac{13}{9}\) or \(-1 \frac{4}{9}\).

Step 1: Identify a Common Denominator

To perform the subtraction of fractions, we need to find a common denominator. The denominators here are 3 and 9. The smallest common multiple of 3 and 9 is 9. So, our common denominator will be 9.

Step 2: Convert Fractions to a Common Denominator

Convert both fractions so that they share the common denominator of 9. For the first fraction: \(-\frac{2}{3} = -\frac{2 \times 3}{3 \times 3} = -\frac{6}{9}\).The second fraction is already expressed with a denominator of 9: \(-\frac{7}{9}\).

Step 3: Subtract the Fractions

Now subtract the second fraction from the first (after conversion):\[ -\frac{6}{9} -\frac{7}{9} = -\left(\frac{6}{9} + \frac{7}{9}\right) = -\frac{13}{9} \].

Step 4: Simplify the Result if Necessary

The resulting fraction \(-\frac{13}{9}\) is already in its simplest form, but it can be expressed as a mixed number:\(-\frac{13}{9} = -1 \frac{4}{9}\).

Key Concepts

Fractions

Fractions are representations of parts of a whole. They consist of two numbers: the numerator (the top part) and the denominator (the bottom part). In the fraction \(-\frac{2}{3}\), \(-2\) is the numerator, indicating the part we have, while \(3\) is the denominator, showing into how many equal parts the whole is divided.
When dealing with negative fractions, like in the given problem, the negative sign can be placed either in front of the fraction, with the numerator, or with the denominator. This means \(-\frac{2}{3}\) is equivalent to \(\frac{-2}{3}\) or \(\frac{2}{-3}\).
Fractions can be added, subtracted, multiplied, or divided, but there are specific rules associated with these operations, especially involving subtraction.

Common Denominator

Before we can subtract fractions, it's crucial each fraction has the same denominator, known as a 'common denominator'.
In our exercise, the fractions \(-\frac{2}{3}\) and \(-\frac{7}{9}\) needed a common denominator to properly perform the subtraction. Finding the lowest common denominator involves identifying the least common multiple (LCM) of the denominators present—which were 3 and 9 in this case.
The smallest common multiple of 3 and 9 is 9, so we adjust the fraction \(-\frac{2}{3}\) by multiplying both numerator and denominator by 3, resulting in \(-\frac{6}{9}\), ensuring both fractions share the denominator of 9 while maintaining equivalent values.

Subtraction of Fractions

Subtracting fractions involves taking one fraction away from another, once you have a common denominator. This means we perform the subtraction on the numerators, keeping the common denominator unchanged.
For example, in the exercise involving the fractions \(-\frac{6}{9}\) and \(-\frac{7}{9}\), we subtract \(6\) from \(7\), yielding \(-13\) over the common denominator of 9, resulting in \(-\frac{13}{9}\).
After performing the subtraction, ensure the result is in its simplest form, meaning there are no common factors between the numerator and the denominator. If necessary, express the result as a mixed number, such as converting \(-\frac{13}{9}\) to \(-1 \frac{4}{9}\), which involves dividing the numerator by the denominator to find the whole number part, and using the remainder as the new numerator.