Problem 32 - Prealgebra

Question

Find sum or difference. Write in simplest form. \(12 \frac{2}{6}-13 \frac{3}{6}\)

Step-by-Step Solution

Answer

The simplest form of the expression is \(-1 \frac{1}{6}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For the first number, \(12 \frac{2}{6}\), the improper fraction is \(\frac{(12 \times 6) + 2}{6} = \frac{74}{6}\). For the second number, \(13 \frac{3}{6}\), the improper fraction is \(\frac{(13 \times 6) + 3}{6} = \frac{81}{6}\).

Step 2: Subtract Improper Fractions

Subtract the two improper fractions: \(\frac{74}{6} - \frac{81}{6} = \frac{74 - 81}{6} = \frac{-7}{6}\).

Step 3: Simplify the Result

The fraction \(\frac{-7}{6}\) is already in its simplest form since 7 and 6 have no common factors other than 1.

Step 4: Final Step: Write the Answer in Mixed Number Form

Convert \(-\frac{7}{6}\) back to a mixed number, which is \(-1 \frac{1}{6}\), since \(-6/6 = -1\) and \(-1\) remains as the whole number part, with \(-1/6\) as the fraction part.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used when the fraction alone could be challenging to interpret or manage. For example, in the number \(12\frac{2}{6}\), 12 is the whole number, and \(\frac{2}{6}\) is the fraction part. To effectively work with mixed numbers, especially when performing operations like addition or subtraction, it's often helpful to convert them to improper fractions. This conversion makes calculations easier and more straightforward.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. Converting a mixed number to an improper fraction involves a few simple steps:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place this result over the original denominator.

For example, converting \(12\frac{2}{6}\) into an improper fraction involves multiplying 12 by 6 and adding 2, which gives 74. Therefore, the improper fraction is \(\frac{74}{6}\). Similarly, \(13\frac{3}{6}\) becomes \(\frac{81}{6}\). This step is crucial for performing arithmetic operations like subtraction.

Simplest Form

Simplifying a fraction means reducing it to its smallest possible numerator and denominator while still retaining its original value. To simplify \(\frac{-7}{6}\), you check for any common factors between the numerator and the denominator. In this case, 7 and 6 have no common factors other than 1. Therefore, \(\frac{-7}{6}\) is already in its simplest form. Always seek to simplify your final answer, as it is considered best practice in mathematics. A simplified fraction is clearer and often easier to work with in subsequent calculations.

Subtraction of Fractions

Subtracting fractions requires common denominators. Fractions with the same denominator, like \(\frac{74}{6}\) and \(\frac{81}{6}\), are subtracted by simply subtracting their numerators:

• Subtract 81 from 74, which results in \(-7\).
• Place the result, \(-7\), over the common denominator, 6.

The result is \(\frac{-7}{6}\). If dealing with unlike denominators, this requires finding a common denominator, typically the least common multiple. But in our case, both fractions already shared the denominator, which made subtraction straightforward.