Problem 4 - Elementary Algebra

Question

Add or subtract as indicated, and express your answers in lowest terms. (Objective 1) $$\frac{11}{13}-\frac{6}{13}$$

Step-by-Step Solution

Answer

The answer is \( \frac{5}{13} \).

Step 1: Identify the Operation and Denominators

The problem asks us to subtract two fractions: \( \frac{11}{13} \) and \( \frac{6}{13} \). Both fractions have the same denominator, which is 13. Since the denominators are the same, we can perform the subtraction directly on the numerators.

Step 2: Subtract the Numerators

Subtract the numerators of the fractions: \( 11 - 6 \). This gives us \( 5 \). Therefore, the result of the subtraction is \( \frac{5}{13} \).

Step 3: Simplify the Fraction

Check if the resulting fraction \( \frac{5}{13} \) can be simplified. Since 5 and 13 have no common factors other than 1, \( \frac{5}{13} \) is already in its lowest terms.

Key Concepts

Understanding Fractions

Fractions are a way of expressing parts of a whole. They consist of two numbers: the numerator and the denominator. The numerator, displayed above the line, represents how many parts of the whole we have. The denominator, found below the line, indicates the total number of equal parts the whole is divided into. Picture a pie cut into 13 slices; if you have 11 slices, you can represent this as the fraction \(\frac{11}{13}\).

• Denominators must be the same to add or subtract fractions directly.
• If the denominators are different, a common denominator must be found.

For the operation \(\frac{11}{13} - \frac{6}{13}\), the denominators are already the same, so subtraction can occur directly on the numerators.

Role of Numerators

The numerator plays a crucial role in fraction operations like subtraction. It is the top number of a fraction and indicates how many parts we are focusing on. In the given problem, we subtract the numerators of the two fractions because their denominators are already identical.

• To subtract \(\frac{11}{13} - \frac{6}{13}\), subtract the numerators: 11 minus 6 equals 5.
• This gives a resulting fraction of \(\frac{5}{13}\).

Remember, only the numerators change during subtraction when denominators are the same. The denominator remains unchanged, ensuring the fractions are parts of the same whole.

Lowest Terms Simplification

Once you have subtracted the fractions, it is crucial to check if your answer can be reduced to its simplest form, known as lowest terms. This means simplifying the fraction so that the numerator and the denominator have no common divisors other than 1.
To determine whether a fraction is in lowest terms:

• Check if any number other than 1 divides both the numerator and the denominator evenly.
• If no such number exists, the fraction is already in its simplest form.

For \(\frac{5}{13}\), 5 and 13 have no common factors other than 1, confirming that \(\frac{5}{13}\) is indeed in its lowest terms.
Working in lowest terms simplifies expressions and ensures a consistent, streamlined approach to dealing with fractions.