Problem 40 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(-2 \frac{1}{6} y+8 \frac{5}{6} y\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{20}{3} y\).

Step 1: Express Mixed Numbers as Improper Fractions

First, we need to convert the mixed numbers into improper fractions. \(-2 \frac{1}{6}\) is equal to \(-\frac{13}{6}\) and \(8 \frac{5}{6}\) is equal to \(\frac{53}{6}\).

Step 2: Set Up the Expression with Fractions

Now, write the expression with fractions: \(-\frac{13}{6} y + \frac{53}{6} y\).

Step 3: Combine Like Terms

Since the fractions have the same denominator, we can combine them easily: \(\left(-\frac{13}{6} + \frac{53}{6}\right) y\).

Step 4: Perform Fraction Addition

Add the fractions: \(-\frac{13}{6} + \frac{53}{6} = \frac{40}{6}\).

Step 5: Simplify the Fraction

Simplify \(\frac{40}{6}\) by dividing both the numerator and the denominator by 2: \(\frac{40}{6} = \frac{20}{3}\).

Step 6: Final Expression

The simplified expression is \(\frac{20}{3} y\).

Key Concepts

Mixed Numbers

Mixed numbers are values that combine a whole number with a fraction. They are common in everyday contexts, such as measurements in cooking or construction. Understanding mixed numbers is essential for converting them into a form we can easily work with in algebra.
To convert a mixed number into an improper fraction, you multiply the whole number by the denominator of the fraction, then add the numerator.

• For example, in the number \[-2 \frac{1}{6}\], multiply \(-2\) by \(6\) to get \(-12\).
• Add the numerator \(1\), resulting in \(-13\).
• Place \(-13\) over the original denominator \(6\), to form \(-\frac{13}{6}\).

This method is incredibly useful when performing operations like addition or subtraction on mixed numbers.

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This state indicates that the value is greater than or equal to one whole. Improper fractions like \(-\frac{13}{6}\) are often easier to work with in algebraic calculations because they don't mix parts of whole numbers with fractions.
To convert back from an improper fraction to a mixed number:

• Divide the numerator by the denominator. The result provides the whole number.
• The remainder becomes the new numerator.
• For example, converting \(-\frac{13}{6}\) back to a mixed number involves dividing \(-13\) by \(6\). The quotient \(-2\) is the whole number, and the remainder \(1\) is the numerator of the fraction, resulting in \(-2 \frac{1}{6}\).

Working with improper fractions simplifies the processes of algebraic manipulation, like addition and subtraction.

Fraction Addition

Adding fractions involves finding common denominators to combine the values accurately. In algebra, when dealing with fractions like those from converted mixed numbers, the process remains the same as traditional fraction addition.
Steps to follow for adding fractions:

• Ensure both fractions have the same denominator. With \(-\frac{13}{6}\) and \(\frac{53}{6}\), both have the same denominator of \(6\), so no further adjustment is required.
• Combine the numerators while keeping the common denominator, resulting in a new numerator. Here, \(-13 + 53 = 40\), maintaining the denominator of \(6\), forms \(\frac{40}{6}\).

The requirement for a common denominator ensures accuracy in the sum, simplifying the process of solving algebraic fractions.

Simplifying Fractions

Simplifying a fraction involves reducing it to its most minimal form, where the numerator and denominator share no common factors other than one. This step is crucial for clarity and precision in mathematical solutions.
To simplify the fraction \(\frac{40}{6}\):

• Identify the greatest common factor (GCF) of the numerator and denominator. Here, the GCF is \(2\).
• Divide both the numerator and denominator by \(2\). This results in \(\frac{20}{3}\).

Simplifying fractions is an essential skill that ensures the expression is tidy, concise, and understandable.