Problem 103 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{11}{21}-\frac{8}{21} $$

Step-by-Step Solution

Answer

\( \frac{11}{21} - \frac{8}{21} = \frac{1}{7} \) after simplification.

Step 1: Understanding the Problem

We need to perform the subtraction operation on fractions with like denominators. The problem is:\[ \frac{11}{21} - \frac{8}{21} \] The denominators are the same (21), which allows us to directly subtract the numerators.

Step 2: Subtract the Numerators

Since the fractions have the same denominator, subtract the numerators directly:\[ 11 - 8 = 3 \] Thus, the fraction becomes:\[ \frac{3}{21} \]

Step 3: Simplify the Fraction

Now, simplify the fraction \( \frac{3}{21} \). Find the greatest common divisor (GCD) of 3 and 21, which is 3. Divide both the numerator and denominator by their GCD:\[ \frac{3 \div 3}{21 \div 3} = \frac{1}{7} \] So, the simplified fraction is \( \frac{1}{7} \).

Key Concepts

Like Denominators

When performing fraction subtraction, like denominators are key. This means that both fractions you want to subtract have the exact same number at the bottom, which is the denominator. For instance, in the operation \( \frac{11}{21} - \frac{8}{21} \), both fractions share 21 as the denominator. This similarity in the denominators allows us to straightforwardly subtract the numerators, the numbers at the top.

Having like denominators simplifies arithmetic operations between fractions, and you don’t have to deal with making the denominators the same. Thus, you can proceed directly to work with the numerators—it's efficient and often simplifies problems significantly. Remember, if denominators are not alike, you’ll need to find a common denominator first, which can be more complex.

Simplifying Fractions

Once you've performed the initial subtraction in a fraction problem, simplifying the result is an important next step. Simplifying a fraction means finding an equivalent fraction in its simplest form, where the numerator and denominator have no common factors other than 1.

In our example of \( \frac{3}{21} \), both 3 and 21 share a factor of 3. Thus, we divide them by 3 to simplify. This reduces the fraction to \( \frac{1}{7} \), which is a more straightforward, simpler representation. Simplifying not only makes fractions easier to work with but also helps in better understanding and communicating the result. Repeated practice will improve your ability to spot when fractions can be simplified.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is a fundamental concept in simplifying fractions. It is the largest number that divides both the numerator and denominator without leaving a remainder. Knowing how to determine the GCD can simplify fractions more smoothly.

To find the GCD, list out the factors of both numbers. In \( \frac{3}{21} \), the factors of 3 are 1 and 3, while the factors of 21 are 1, 3, 7, and 21. The highest number common to both lists is 3, which makes it the GCD. By dividing the numerator and denominator by the GCD, \( \frac{3}{3} = 1 \) and \( \frac{21}{3} = 7 \), you achieve the simplest form, \( \frac{1}{7} \).

• The GCD streamlines the process of reducing fractions.
• It also ensures calculations remain accurate and legible.

Understanding the GCD is not only crucial for simplifying fractions but is useful in various areas of math.