Problem 15 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(6 \frac{1}{3}-4 \frac{1}{4}\)

Step-by-Step Solution

Answer

The result of the subtraction is \(2 \frac{1}{12}\).

Step 1: Convert to Improper Fractions

First, we need to convert each mixed number into an improper fraction. For the first number, \(6 \frac{1}{3}\), multiply 6 by 3 and add 1: \((6 \times 3) + 1 = 19\). So, \(6 \frac{1}{3} = \frac{19}{3}\). For the second number, \(4 \frac{1}{4}\), multiply 4 by 4 and add 1: \((4 \times 4) + 1 = 17\). Thus, \(4 \frac{1}{4} = \frac{17}{4}\).

Step 2: Find a Common Denominator

To subtract these fractions, we need a common denominator. The denominators are 3 and 4, so the least common multiple is 12. Convert \(\frac{19}{3}\) and \(\frac{17}{4}\) to fractions with a denominator of 12. For \(\frac{19}{3}\), multiply the numerator and denominator by 4: \(\frac{19 \times 4}{3 \times 4} = \frac{76}{12}\). For \(\frac{17}{4}\), multiply the numerator and denominator by 3: \(\frac{17 \times 3}{4 \times 3} = \frac{51}{12}\).

Step 3: Subtract the Fractions

Now subtract the fractions: \(\frac{76}{12} - \frac{51}{12} = \frac{25}{12}\). Subtracting directly, we get \(\frac{25}{12}\).

Step 4: Convert to a Mixed Number

Convert \(\frac{25}{12}\) back into a mixed number. Divide 25 by 12, which goes 2 times with a remainder of 1. So, \(\frac{25}{12} = 2 \frac{1}{12}\).

Key Concepts

Improper Fractions

Improper fractions can seem tricky at first, but they actually make operations like addition and subtraction quite manageable. An improper fraction is a fraction where the numerator, the top part, is greater than or equal to the denominator, the bottom part. This can occur when we convert mixed numbers into fractions that express the same value but are easier to work with in arithmetic calculations.

For example, if you have a mixed number like \(6 \frac{1}{3}\), here's how you convert it: multiply the whole number (6) by the denominator of the fraction (3), then add the numerator (1). So, \((6 \times 3) + 1 = 19\). Thus, the mixed number \(6 \frac{1}{3}\) becomes \(\frac{19}{3}\).

This step ensures that both numbers in your operation are in the same format, which simplifies the process of finding a common denominator and subtraction.

Common Denominator

When subtracting fractions, having a common denominator is crucial. A common denominator is a shared multiple of the denominators of two or more fractions. Using a common denominator allows the fractions to be compared or subtracted because it ensures both fractions are in terms of the same whole.

To find a common denominator, you need the least common multiple (LCM) of the original denominators. In the exercise, the denominators were 3 and 4, with an LCM of 12. Once identified, adjust each fraction so that their denominators equal this LCM.

• For \(\frac{19}{3}\), multiply both numerator and denominator by 4 to get \(\frac{76}{12}\).
• For \(\frac{17}{4}\), multiply both numerator and denominator by 3 to obtain \(\frac{51}{12}\).

Now both fractions have a denominator of 12, and subtraction becomes straightforward.

Subtracting Fractions

Subtracting fractions with a common denominator is a straightforward process. Once the fractions have the same denominator, you simply subtract the numerators and place the result over the common denominator.

For example, with the fractions \(\frac{76}{12}\) and \(\frac{51}{12}\), subtract the numerators: \(76 - 51 = 25\). This gives us \(\frac{25}{12}\).

Finally, the fraction \(\frac{25}{12}\) can be converted back into a mixed number for easier interpretation. Divide the numerator by the denominator: 25 divided by 12 is 2 with a remainder of 1. Therefore, \(\frac{25}{12}\) is equivalent to \(2 \frac{1}{12}\). This ensures the result is clear and easy to understand.