Problem 57 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{23}{105}+\frac{4}{105} $$

Step-by-Step Solution

Answer

The result is \(\frac{9}{35}\).

Step 1: Identify the Denominators

First, observe that the denominators of both fractions are the same: 105. This means we can add them directly, as we do not need to find a common denominator.

Step 2: Add the Numerators

Since the denominators are identical, we add the numerators: \(23 + 4 = 27\). This gives us the new fraction: \(\frac{27}{105}\).

Step 3: Simplify the Fraction

Next, we simplify the fraction \(\frac{27}{105}\) by finding the greatest common divisor (GCD) of the numbers 27 and 105. The GCD is 3. Divide both the numerator and the denominator by 3 to simplify: \(\frac{27\div 3}{105\div 3} = \frac{9}{35}\).

Step 4: Verify the Simplification

Check if \(9\) and \(35\) have any common divisors other than 1. Since they do not, \(\frac{9}{35}\) is in its simplest form.

Key Concepts

Common Denominator

When adding fractions, finding a common denominator is usually a crucial first step. The denominator is the number below the fraction line, and having the same denominator means that we can add the fractions directly. By sharing a common unit (or base), the fractions become easier to combine. In the example given, both fractions \( \frac{23}{105} \) and \( \frac{4}{105} \) already share the same denominator, which is 105. This means we can move directly to adding the numerators without any extra steps.

• Check the denominators of each fraction.
• If they are the same, you can proceed to the next step.
• If they are different, find the least common multiple to have a common denominator.

When denominators are different, we adjust the fractions so that they share the same base before proceeding to addition.

Numerator Addition

Once common denominators are established, adding the fractions is straightforward. The numerators, or the top numbers, dictate how many parts we have in total. In our exercise, since the denominators were already the same, the next task was to add the numerators directly.
Let's break it down:

• The first step is to simply add the numerators of the fractions.
• For instance, with \( \frac{23}{105} + \frac{4}{105} \), the numerators are 23 and 4.

Thus, combining these yields a numerator of 27, resulting in the fraction \( \frac{27}{105} \). This step emphasizes addition without altering the denominator until the result is achieved.
Fractions with consistent denominators ensure that we only need to focus on summing up the numerators, simplifying the calculation process.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest or smallest form. This is important for clarity and simplicity, ensuring that the numbers are as reduced as possible. Once we have the basic fraction after addition, simplifying becomes the focus.
In the example with \( \frac{27}{105} \):

• First, identify if both the numerator (27) and the denominator (105) share any common factors.
• Use the greatest common divisor (GCD) to divide both terms.

After engaging this procedure, simplifying with the GCD helps us reach the fraction \( \frac{9}{35} \). The goal of simplification is always to make complex fractions more manageable without changing their value.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor, is crucial for fraction simplification. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Determining the GCD allows us to reduce fractions to their simplest form.
For the fraction \( \frac{27}{105} \):

• First, list the factors of 27 and 105.
• For 27, the factors are 1, 3, 9, and 27.
• For 105, they are 1, 3, 5, 7, 15, 21, 35, and 105.

The highest common factor is 3. This means that when we divide both the numerator and the denominator by 3, we achieve the simplest form, \( \frac{9}{35} \).Using the GCD not only simplifies fractions but also establishes the efficiency of mathematical operations.