Problem 23 - Intermediate Algebra

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 into improper fractions to make the calculation easier. \(4 \frac{1}{3}\) becomes \(\frac{13}{3}\) because \(4 \times 3 + 1 = 13\). For \(-1 \frac{1}{6}\), it becomes \(-\frac{7}{6}\) because \(1 \times 6 + 1 = 7\).

Step 2: Rewrite the Subtraction Problem

The problem now is: \(\frac{13}{3} - (-\frac{7}{6})\). According to the rules of arithmetic, subtracting a negative is the same as adding a positive, so this can be rewritten as \(\frac{13}{3} + \frac{7}{6}\).

Step 3: Find a Common Denominator

The denominators here are 3 and 6. The least common denominator (LCD) for these numbers is 6. Rewrite \(\frac{13}{3}\) as a fraction with 6 as the denominator: \(\frac{13}{3} = \frac{26}{6}\).

Step 4: Add the Fractions

Now that they have the same denominator, add the fractions: \(\frac{26}{6} + \frac{7}{6} = \frac{33}{6}\).

Step 5: Simplify the Result

Finally, simplify \(\frac{33}{6}\). Divide the numerator and the denominator by their greatest common divisor, which is 3: \(\frac{33}{6} = \frac{11}{2}\). This can also be expressed as the mixed number \(5 \frac{1}{2}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are used often in everyday life, for example, when measuring ingredients or talking about time and distances. But to perform mathematical operations like addition or subtraction, it is easier to convert them into improper fractions.
To convert a mixed number to an improper fraction:

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

For instance, converting the mixed number \(4 \frac{1}{3}\) involves multiplying 4 (the whole number) by 3 (the denominator), resulting in 12. Then, add 1 (the numerator) to get 13. Therefore, \(4 \frac{1}{3}\) becomes \(\frac{13}{3}\). This method helps make calculations more straightforward.

Improper Fractions

Improper fractions have numerators that are larger than their denominators. Unlike mixed numbers, they may initially seem awkward, but they're extremely useful in arithmetic operations.
Working with improper fractions allows for smoother calculations because they eliminate the need to deal with separate whole numbers and fractional parts.
Let's use \(-1 \frac{1}{6}\) as an example. To convert this into an improper fraction:

• Multiply the whole number (1) by the denominator (6), giving you 6.
• Add the numerator (1) to obtain 7.
• Simplify into the negative form: \(-\frac{7}{6}\).

Now, when these fractions are improper, they make working with larger calculations like finding common denominators very manageable.

Common Denominator

When adding or subtracting fractions, it is essential that they share a common denominator. This allows for the straightforward addition or subtraction of numerators. The least common denominator (LCD) is the smallest number that both denominators can evenly divide into, making it efficient for calculations.
For the fractions \(\frac{13}{3}\) and \(\frac{7}{6}\), the denominators are 3 and 6. The least common denominator is 6.
Converting \(\frac{13}{3}\) to a fraction with a denominator of 6 involves:

• Finding the factor you multiply 3 by to get 6, which is 2.
• Multiply both the numerator and the denominator by this factor. Thus, \(\frac{13}{3}\) becomes \(\frac{26}{6}\).

With this setup, both fractions now have a common denominator, making further operations straightforward.

Simplifying Fractions

After performing operations with fractions, like addition or subtraction, the next step often involves simplifying the result. Simplification makes the fraction easier to read and understand by reducing it to its lowest terms.
To simplify a fraction:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by the GCD.

Consider \(\frac{33}{6}\). The GCD of 33 and 6 is 3. Dividing both by 3 gives us \(\frac{11}{2}\).
This fraction could also be expressed as the mixed number \(5 \frac{1}{2}\) if desired, but both forms are valid representations of the same value.