Problem 81 - Prealgebra

Question

Add or subtract as indicated. $$\frac{1}{6}-\frac{4}{3}$$

Step-by-Step Solution

Answer

The difference is \( \frac{-7}{6} \).

Step 1: Identify the Operation

The operation required in the exercise is subtraction: \( \frac{1}{6} - \frac{4}{3} \). Our goal is to subtract the second fraction from the first.

Step 2: Find a Common Denominator

To perform subtraction, both fractions must have the same denominator. The denominators are 6 and 3. The least common denominator (LCD) of 6 and 3 is 6.

Step 3: Convert Fractions to Common Denominator

Convert the fraction with a different denominator to the common denominator. \( \frac{4}{3} \) should be converted: \( \frac{4}{3} = \frac{4 \times 2}{3 \times 2} = \frac{8}{6} \). So, our new equation is \( \frac{1}{6} - \frac{8}{6} \).

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \( \frac{1}{6} - \frac{8}{6} = \frac{1 - 8}{6} = \frac{-7}{6} \).

Step 5: Simplify the Result

Check if the resulting fraction can be simplified. \( \frac{-7}{6} \) is already in its simplest form. The result is a negative improper fraction.

Key Concepts

Common Denominator

When dealing with fractions, especially when adding or subtracting them, finding a common denominator is essential. A common denominator is a shared multiple of the denominators of two fractions. This step allows us to rewrite the fractions so that they have the same base below the fraction bar, making them easier to compare or operate on.

To find a common denominator, look for the least common multiple of the denominators. In our example problem of subtracting \( \frac{1}{6} \) and \( \frac{4}{3} \), the denominators are 6 and 3. The least common multiple of these numbers is 6, which means that both fractions can be expressed with 6 as a common denominator:

• \( \frac{1}{6} \) already has the denominator of 6, so it remains as it is.
• For \( \frac{4}{3} \), multiply both the numerator and denominator by 2 to convert it to \( \frac{8}{6} \).

Finding a common denominator makes the arithmetic straightforward, as you can now focus on the numerators only. This is a fundamental step in manipulating fractions.

Improper Fraction

An improper fraction occurs when the numerator is larger than the denominator, indicating that the fraction is larger than one whole. In our exercise, after performing the subtraction, we ended with \( \frac{-7}{6} \), which is an improper fraction.

Improper fractions might seem a bit intimidating at first, but they are very useful when dealing with arithmetic involving fractions. It's important to recognize them because they tell us more about the quantity and how it compares to whole numbers.

• An improper fraction such as \( \frac{-7}{6} \) is also negative, which suggests that its value is less than zero.
• In some situations, you might want to convert improper fractions into mixed numbers for easier interpretation, or simply understand it as "how many whole parts" plus "what part of the whole".

Understanding improper fractions helps in quickly assessing the size and relativity of fractions compared to whole numbers.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, where the numerator and the denominator share no common factors other than 1. This process makes the fraction easier to understand and use.

In our example, after subtracting, we got \( \frac{-7}{6} \). When simplifying fractions, the first step is to check if the numerator and denominator share any common factors. If they do, you divide both by the greatest common factor:

• In \( \frac{-7}{6} \), 7 and 6 are co-prime, since 7 is a prime number and does not divide 6. Thus, it can't be simplified further.

Sometimes it might be tempting to keep simplifying or transforming fractions, but in cases where the simplest form is already reached, like \( -\frac{7}{6} \), your work is done! Clear and simple fractions are less daunting and serve as a solid foundation for more complex calculations.