Problem 77 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \begin{aligned} &\frac{23}{105}-\frac{2}{105} \end{aligned} $$

Step-by-Step Solution

Answer

\( \frac{1}{5} \)

Step 1: Identify the Denominators

In this subtraction problem, both fractions have the same denominator: 105. This makes the operation straightforward since the denominators do not need to be adjusted.

Step 2: Subtract the Numerators

Since the denominators are equal, simply subtract the numerators. So we calculate: \[23 - 2 = 21\].

Step 3: Write the Resulting Fraction

After subtracting the numerators, the resulting fraction is \( \frac{21}{105} \).

Step 4: Simplify the Fraction

To simplify \( \frac{21}{105} \), find the greatest common divisor (GCD) of 21 and 105. The GCD is 21. Divide both the numerator and the denominator by their GCD: \( \frac{21}{21} = 1 \) and \( \frac{105}{21} = 5 \). So the simplified fraction is \( \frac{1}{5} \).

Key Concepts

Simplifying Fractions

To simplify a fraction means to make it as simple as possible. Specifically, this involves reducing the fraction so that there are no common factors between the numerator (the top number) and the denominator (the bottom number), except for 1.
This process helps to express the fraction in the smallest possible terms but with the same value.Here is how you simplify a fraction:

• Find the Greatest Common Divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

Let's take the fraction \( \frac{21}{105} \) as our example. First, we determine the GCD, which is 21 in this case. Then, we divide both the numerator and the denominator by 21:

• Numerator: \( 21 \div 21 = 1 \)
• Denominator: \( 105 \div 21 = 5 \)

Thus, \( \frac{21}{105} \) simplifies to \( \frac{1}{5} \). This tells us that \( \frac{21}{105} \) is equivalent to \( \frac{1}{5} \), but \( \frac{1}{5} \) is easier to use and understand.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCD is a vital step in simplifying fractions. You can determine the GCD through several methods:

• Prime Factorization: Break down each number into prime factors and then multiply the smallest powers of common factors.
• Euclidean Algorithm: This method involves repeated division and finding remainders. It's efficient, especially for larger numbers.

For instance, consider the numbers 21 and 105 again:
- **Prime Factorization**: - 21 is \( 3 \times 7 \) - 105 is \( 3 \times 5 \times 7 \) Both numbers share the factors 3 and 7, so the GCD is \( 3 \times 7 = 21 \).Knowing how to find the GCD will make simplifying fractions quick and easy, aiding your ability to deal with complex fraction operations.

Subtracting Fractions

Subtracting fractions involves dealing with the numerators, given the denominators are the same. This is a straightforward process if both fractions share the same denominator, as it simplifies the calculation.To subtract fractions:

• Ensure the fractions have a common denominator.
• Subtract the numerators while keeping the denominator the same.

For example, to subtract \( \frac{23}{105} \) from \( \frac{2}{105} \):

• The denominators are both 105, so we don't need to find a common denominator.
• Proceed by subtracting the numerators: \( 23 - 2 = 21 \).

After performing this subtraction, you get the fraction \( \frac{21}{105} \). Since subtracting fractions with the same denominator only involves their numerators, understanding this process helps simplify many problems. Just remember to simplify the fraction afterwards!