Problem 40 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ -\frac{4}{9}-\frac{2}{9}-\frac{5}{9} $$

Step-by-Step Solution

Answer

-1 and 2/9

Step 1: Adding Fractions with the Same Denominator

When adding or subtracting fractions with the same denominator, combine the numerators and keep the same denominator. So, it would be (-4-2-5)/9.

Step 2: Combine the Numerators

Combine the numerators. Now it is \(-\frac{11}{9}\).

Step 3: Express the Fraction as Mixed Number

If the numerator is greater than the denominator, the fraction can be expressed as a mixed number. Since 11/9 > 1, this can be expressed as \(1\frac{2}{9}\). Because it is negative, the result is \(-1\frac{2}{9}\).

Key Concepts

Adding Fractions

Fractions are a way to represent parts of a whole. When adding fractions, the key factor is the denominator, which determines the size of each part. To add fractions with the same denominator, follow a straightforward process:

• Ensure the denominators are the same. If they aren't, this usually means you'll need to find a common denominator.
• Add or subtract the numerators, the top numbers, while keeping the denominator the same.

For example, in the expression\[ -\frac{4}{9} - \frac{2}{9} - \frac{5}{9} \]the denominator is 9 for each fraction. Simply focus on the numerators and subtract them:\[ (-4) + (-2) + (-5) = -11 \]Then place the result over the common denominator:\[ -\frac{11}{9} \]Maintaining the same denominator ensures consistency in the size of parts being added or subtracted.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They appear when a fraction's numerator is larger than its denominator, indicating a value greater than 1. Converting an improper fraction to a mixed number requires a simple division:

• Divide the numerator by the denominator to find the whole number.
• The remainder becomes the numerator of the fraction part, while the denominator remains unchanged.

For example, with the fraction \( -\frac{11}{9} \),divide 11 by 9:- 11 goes into 9 once, making 1 the whole number part.- The remainder is 2, forming the new fraction \( \frac{2}{9} \).Thus, the improper fraction becomes the mixed number:\( 1\frac{2}{9} \).Since the original fraction was negative, so is the mixed number, resulting in \( -1\frac{2}{9} \). Mixed numbers provide a more intuitive way to understand quantities that exceed whole parts.

Numerators and Denominators

Understanding the roles of numerators and denominators in fractions is crucial. The numerator, the top part, represents how many parts of the whole are being considered. The denominator, the bottom part, indicates the total number of equal parts the whole is divided into.

• The numerator dictates the quantity or count of parts.
• The denominator sets the size of each part, maintaining uniformity across the fractions being compared or combined.

In \( \frac{4}{9} \):- The numerator 4 implies four parts are taken.- The denominator 9 signifies the whole is split into nine equal segments.Understanding these components helps in adding, subtracting, or comparing fractions. In operations, such as addition or subtraction, ensuring fraction denominators are identical allows direct manipulation of numerators without altering the base size of the parts.