Problem 19 - Algebra for College Students

Question

Perform the following operations with real numbers. $$4 \frac{1}{3}-\left(-1 \frac{1}{6}\right)$$

Step-by-Step Solution

Answer

The result is \(5 \frac{1}{2}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. The fraction \(4 \frac{1}{3}\) is converted to \(\frac{13}{3}\) by multiplying 4 by 3 and adding 1. Likewise, the fraction \(1 \frac{1}{6}\) is \(\frac{7}{6}\) by multiplying 1 by 6 and adding 1.

Step 2: Rewrite Subtraction as Addition of Opposites

Rewrite the subtraction expression \(4 \frac{1}{3} - (-1 \frac{1}{6})\) as the addition of the opposite. This is equivalent to \(4 \frac{1}{3} + 1 \frac{1}{6}\).

Step 3: Convert Again to Improper Fractions (if needed)

Ensure the fractions are improper: \(\frac{13}{3}\) and \(\frac{7}{6}\). These were converted in the first step.

Step 4: Find a Common Denominator

Find a common denominator for the fractions \(\frac{13}{3}\) and \(\frac{7}{6}\). The least common multiple of 3 and 6 is 6, so use 6 as the common denominator.

Step 5: Convert Fractions with Common Denominator

Convert \(\frac{13}{3}\) to a fraction with the denominator of 6, which is \(\frac{26}{6}\). Leave \(\frac{7}{6}\) as it is, since it already has the denominator 6.

Step 6: Add the Fractions

Add the fractions with a common denominator: \(\frac{26}{6} + \frac{7}{6} = \frac{33}{6}\).

Step 7: Simplify the Result

Simplify the fraction \(\frac{33}{6}\). Dividing the numerator and the denominator by their greatest common divisor, which is 3, results in \(\frac{11}{2}\).

Step 8: Convert Back to Mixed Number

Convert \(\frac{11}{2}\) back to a mixed number, which is \(5 \frac{1}{2}\). Divide 11 by 2 to get 5 with a remainder of 1, so the final mixed number is \(5 \frac{1}{2}\).

Key Concepts

Mixed Numbers to Improper Fractions

Sometimes, it's more convenient to work with improper fractions instead of mixed numbers when performing arithmetic operations. A mixed number consists of an integer and a proper fraction, like the example of \(4 \frac{1}{3}\). To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction. For \(4 \frac{1}{3}\), multiply 4 by 3 to get 12.
• Add the numerator of the fraction. Add 1 to 12, resulting in 13.
• Place this result over the original denominator to get \(\frac{13}{3}\).

Repeating this process for \(1 \frac{1}{6}\), multiply 1 by 6 to get 6, then add 1 to get 7, which gives the fraction \(\frac{7}{6}\). This method makes further calculations easier to perform.

Finding Common Denominators

When adding or subtracting fractions, they must share a common denominator. This means the denominator—the bottom number of the fraction—should be the same for each fraction involved in the operation. For instance, if you have \(\frac{13}{3}\) and \(\frac{7}{6}\), find the least common multiple (LCM) of 3 and 6.
The LCM of 3 and 6 is 6, which means it's the smallest number divisible by both 3 and 6. Using 6 as the common denominator enables the fractions to be added easily.
Convert \(\frac{13}{3}\) to \(\frac{26}{6}\) so both fractions have the same denominator, making them ready for addition.

Adding Fractions

Once fractions share a common denominator, they can be added directly by adding their numerators while keeping the common denominator. This process helps simplify calculations and avoid errors.

• For our example, take \(\frac{26}{6}\) and \(\frac{7}{6}\).
• Add their numerators: 26 + 7 = 33.
• Keep the denominator as 6, resulting in the fraction \(\frac{33}{6}\).

This combined fraction represents the sum of the original parts since adding fractions collectively means summing up equal parts of a whole.

Simplifying Fractions

After performing addition, the resulting fraction should be simplified to its simplest form. Simplifying involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number. In our example, the fraction \(\frac{33}{6}\) can be simplified:

• Determine the GCD of 33 and 6, which is 3.
• Divide both the numerator and the denominator by this GCD: \(\frac{33}{3} = 11\) and \(\frac{6}{3} = 2\).
• The simplified fraction is \(\frac{11}{2}\).

Optionally, convert this improper fraction back to a mixed number. Since 11 divided by 2 is 5 with a remainder of 1, the mixed number is \(5 \frac{1}{2}\). This step ensures the result is in the most understandable and useful form.