Problem 45 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find a Common Denominator

To perform the operation with fractions, we need a common denominator. The denominators in the fractions are 3 and 9. The least common multiple of 3 and 9 is 9, so that will be our common denominator.

Step 2: Convert the First Fraction

Convert \(-\frac{2}{3}\) to a fraction with denominator 9. Multiply both the numerator and the denominator by 3 to get: \(-\frac{2\times3}{3\times3} = -\frac{6}{9}\).

Step 3: Perform the Subtraction

Now that both fractions have the same denominator, subtract the fractions: \(-\frac{6}{9} - \frac{7}{9} = \frac{-6 - 7}{9} = \frac{-13}{9}\).

Step 4: Simplify if Necessary

The fraction \(-\frac{13}{9}\) is already in its simplest form, as 13 is a prime number and not divisible by 9. Therefore, no further simplification is necessary.

Key Concepts

Fractions

Fractions represent parts of a whole. They consist of a numerator and a denominator.

• The numerator is the top number and indicates how many parts you have.
• The denominator is the bottom number and shows into how many parts the whole is divided.

For example, in the fraction \(-\frac{2}{3}\), 2 is the numerator and 3 is the denominator. Here, the fraction is negative, indicating that you are dealing with a negative quantity. Fractions can be manipulated through various operations like addition, subtraction, multiplication, and division. Working with fractions requires understanding how to maintain the relationship between the numerator and the denominator to represent the same value effectively.

Common Denominator

Finding a common denominator is essential for performing operations such as addition and subtraction on fractions. The common denominator is a shared multiple of the original denominators of the fractions. For the fractions \(-\frac{2}{3}\) and \(-\frac{7}{9}\), you first identify the denominators, which are 3 and 9. The least common multiple of 3 and 9 is 9, making it the ideal common denominator.

• Selecting the least common multiple helps minimize the size of the numbers you work with, simplifying calculations.
• Converting fractions to have this common denominator is crucial for accurately performing mathematical operations.

In this case, it's necessary to adjust \(-\frac{2}{3}\) to \(-\frac{6}{9}\) so that both fractions can be easily combined.

Subtraction

Subtracting fractions requires them to share the same denominator. Once the fractions \(-\frac{6}{9}\) and \(-\frac{7}{9}\) have a common denominator, subtracting the fractions becomes straightforward. To subtract, keep the denominator, and subtract the numerators.

• The calculation becomes: \(-\frac{6}{9} - \frac{7}{9} = \frac{-6 - 7}{9}.\)
• Subtracting the numerators, we get \(-13\).
• The result is \(-\frac{13}{9}\).

This method keeps the value of the fractions correct because only the numerators change, while the denominator stays the same, ensuring that the fractional part is consistently represented.

Simplification

Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In the result from our subtraction, \(-\frac{13}{9}\), both 13 and 9 are analyzed.

• The number 13 is a prime number, meaning it can only be divided by 1 and itself, and it shares no common factors with 9 other than 1.
• This means \(-\frac{13}{9}\) is already in its simplest form.

Simplification is essential when working with fractions, as it provides the cleanest and most visually manageable result, making further operations with them easier to handle.