Problem 46 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$5 \frac{1}{6}-3 \frac{1}{4}$$

Step-by-Step Solution

Answer

The difference is \(1 \frac{11}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(5 \frac{1}{6}\), multiply the whole number 5 by the denominator 6, giving 30, then add the numerator 1: \(\frac{30+1}{6} = \frac{31}{6}\). For \(3 \frac{1}{4}\), multiply 3 by the denominator 4, giving 12, then add the numerator 1: \(\frac{12+1}{4} = \frac{13}{4}\).

Step 2: Find a Common Denominator

To subtract fractions, they must have the same denominator. The denominators here are 6 and 4. The least common multiple of 6 and 4 is 12, so we will use 12 as the common denominator.

Step 3: Convert Fractions to Common Denominator

Convert \(\frac{31}{6}\) to a denominator of 12 by multiplying both the numerator and the denominator by 2: \(\frac{31 \times 2}{6 \times 2} = \frac{62}{12}\). Convert \(\frac{13}{4}\) to a denominator of 12 by multiplying both the numerator and the denominator by 3: \(\frac{13 \times 3}{4 \times 3} = \frac{39}{12}\).

Step 4: Subtract the Fractions

Now subtract the two fractions: \(\frac{62}{12} - \frac{39}{12} = \frac{23}{12}\). Since the denominators are the same, simply subtract the numerators.

Step 5: Convert Improper Fraction to Mixed Number

Convert \(\frac{23}{12}\) back to a mixed number. Divide 23 by 12, which goes 1 time with a remainder of 11. So, \(\frac{23}{12} = 1 \frac{11}{12}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that consist of both a whole number and a fraction combined together. They are used to represent quantities that are more than a whole but not quite another whole number. For example, in the mixed number \(5 \frac{1}{6}\), 5 is the whole number, and \(\frac{1}{6}\) is the fractional part.
To convert a mixed number into an improper fraction, you multiply the whole number by the denominator of the fraction and then add the numerator. This method helps in subtraction or addition of mixed numbers by simplifying the process. So with \(5 \frac{1}{6}\), you multiply 5 by 6 to get 30, add 1, giving \(\frac{31}{6}\). Understanding this conversion is a foundational step in operations involving mixed numbers.

Improper Fractions

Improper fractions have a numerator that is equal to or greater than the denominator. This means that the fraction represents a value that is 1 or greater. For instance, \(\frac{31}{6}\) is an improper fraction because the numerator 31 is greater than the denominator 6.
Starting with mixed numbers, changing them into improper fractions helps to simplify arithmetic operations such as subtraction. Since improper fractions can be easily managed under a common denominator, they are essential in the subtracting process where we need a single unified type of fraction.
Improper fractions can sometimes seem more challenging, but they allow us to sidestep the difficulties that can arise when working directly with mixed numbers.

Least Common Denominator

The least common denominator (LCD) is the smallest multiple that is common to all denominators involved in a problem. In fraction subtraction, such as this exercise, the LCD is crucial.
Finding the LCD makes it possible to rewrite each fraction so they share a common denominator. In our example, we have the fractions \(\frac{31}{6}\) and \(\frac{13}{4}\). Their denominators, 6 and 4, lead us to an LCD of 12, since it's the smallest number both 6 and 4 can divide into evenly.

• Multiply both numerator and denominator by a factor that results in the LCD for each fraction.
• For \(\frac{31}{6}\), multiply by 2 to get \(\frac{62}{12}\).
• For \(\frac{13}{4}\), multiply by 3 to get \(\frac{39}{12}\).

This conversion makes subtraction straightforward as you'll be working with like terms.

Step by Step Solutions

Step-by-step solutions are a methodical way to ensure every part of the calculation is correctly handled, reducing mistakes and making it easier to follow complex processes like the subtraction of fractions involving mixed numbers.
This particular exercise walks you through converting mixed numbers to improper fractions, finding the least common denominator, converting the fractions to a common denominator, performing the subtraction, and finally converting back, if needed, to a mixed number. Here's a simple breakdown:

• Convert mixed numbers to improper fractions.
• Determine the least common denominator.
• Convert to fractions with like denominators.
• Subtract the fractions.
• Convert any result back into a mixed number if applicable.

Following each step carefully ensures clarity and correctness throughout the process. This systematic approach not only provides the correct end result but also strengthens your understanding of each individual mathematical concept involved.