Problem 65 - Intermediate Algebra

Question

Simplify each numerical expression. $$ 4 \frac{1}{12}-\frac{1}{2}\left(\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{47}{12}\).

Step 1: Convert Mixed Number to Improper Fraction

The given expression is \(4 \frac{1}{12} - \frac{1}{2}\left(\frac{1}{3}\right)\). First, convert \(4 \frac{1}{12}\) to an improper fraction. Multiply 4 by 12 and add 1: \(4 \times 12 + 1 = 49\). So the improper fraction is \(\frac{49}{12}\).

Step 2: Simplify the Fraction in Parentheses

The expression involves \(\frac{1}{2}\left(\frac{1}{3}\right)\). Multiply the fractions: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).

Step 3: Subtract the Fractions

Subtract \(\frac{1}{6}\) from \(\frac{49}{12}\). First, find a common denominator. The least common multiple of 12 and 6 is 12. Rewrite \(\frac{1}{6}\) as \(\frac{2}{12}\).

Step 4: Perform the Subtraction

Subtract \(\frac{2}{12}\) from \(\frac{49}{12}\). This gives \(\frac{49}{12} - \frac{2}{12} = \frac{47}{12}\).

Key Concepts

Understanding Improper Fractions

Improper fractions are a common part of mathematics that can appear intimidating at first. However, understanding them is simpler than it seems. An improper fraction is a fraction where the numerator (the number on top) is larger than the denominator (the number below). For example, \( \frac{49}{12} \) is an improper fraction because 49 is greater than 12. These fractions often come from converting mixed numbers. A mixed number, such as \( 4 \frac{1}{12} \), combines a whole number with a proper fraction. Converting it to an improper fraction involves multiplying the whole number by the fraction's denominator and adding the numerator to the result \( (4 \times 12 + 1 = 49) \). This helps when performing operations like addition and subtraction because it eliminates any remaining complexity from the whole number part.

Exploring Least Common Multiple (LCM)

Finding the least common multiple (LCM) is crucial for simplifying expressions involving the addition and subtraction of fractions. The LCM of two numbers is the smallest multiple they both share. When dealing with fractions, the LCM of their denominators becomes the common base needed for operations. For instance, if you need to subtract \(\frac{1}{6}\) from \(\frac{49}{12}\), you should identify the LCM of 12 and 6. Since 12 is a multiple of 6, it serves as the LCM. This allows you to rewrite \(\frac{1}{6}\) with a denominator of 12 as \(\frac{2}{12}\). By establishing a common denominator using the LCM, you can easily perform the subtraction operation on the numerators.

Mastering Fraction Subtraction

Subtracting fractions might seem tricky, but it becomes straightforward with clear steps. First, ensure both fractions share a common denominator; this often involves using the least common multiple (LCM). For example, in our problem, \(\frac{49}{12} - \frac{1}{6}\) requires both fractions to have the same denominator. We convert \(\frac{1}{6}\) to \(\frac{2}{12}\) using the LCM.Once the fractions have the same denominator, subtract the numerators while keeping the denominator unchanged. Here, \(49 - 2 = 47\), resulting in \(\frac{47}{12}\). It's essential to always subtract the numerators only, not the denominators. The process becomes more intuitive with practice, helping tackle a wide range of fraction problems with ease.