Problem 90 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{3}{7}-\frac{2}{5}+\frac{2}{35} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{3}{35} \).

Step 1: Find a Common Denominator

Identify the least common denominator (LCD) for the fractions involved. The fractions are \( \frac{3}{7} \), \( \frac{2}{5} \), and \( \frac{2}{35} \), with denominators 7, 5, and 35, respectively. The LCD is 35, as it is the least common multiple of these numbers.

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the denominator of 35. Thus, \( \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \), \( \frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \), and \( \frac{2}{35} = \frac{2}{35} \) remains the same.

Step 3: Perform the Subtraction and Addition

Subtract \( \frac{14}{35} \) from \( \frac{15}{35} \), which is \( \frac{15-14}{35} = \frac{1}{35} \). Then add \( \frac{2}{35} \), which is \( \frac{1}{35} + \frac{2}{35} = \frac{1+2}{35} = \frac{3}{35} \).

Step 4: Check for Simplification

Verify if \( \frac{3}{35} \) can be simplified. Since there are no common factors between 3 and 35 other than 1, \( \frac{3}{35} \) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

To successfully add or subtract fractions with different denominators, you first need to find a common baseline for all fractions. This baseline is what we call the "least common denominator" (LCD). When dealing with fractions like \( \frac{3}{7} \), \( \frac{2}{5} \), and \( \frac{2}{35} \), determine the LCD by finding the least common multiple (LCM) of the denominators 7, 5, and 35. The LCM is the smallest number that all the denominators divide into evenly.
For these fractions, 35 is the LCM, making it the least common denominator. This is because:

• 35 is a multiple of 7 (7 x 5 = 35).
• 35 is a multiple of 5 (5 x 7 = 35).
• 35 is already the denominator of \( \frac{2}{35} \).

Using the LCD helps streamline the process of working with the fractions by providing a common ground.

Creating Equivalent Fractions

After determining the least common denominator, the next step is to convert each fraction to an equivalent one that shares this new denominator. Equivalent fractions have the same value as the original ones but with the denominator changed to the LCD. This is important when you want to add or subtract fractions horizontally.
For example, transform \( \frac{3}{7} \) into a fraction with 35 as its denominator. Multiply both the numerator and denominator by the number that turns 7 into 35:

• Multiply 3 by 5 to get 15 (\( \frac{3 \times 5}{7 \times 5} = \frac{15}{35} \)).

In the same way, convert \( \frac{2}{5} \) to an equivalent fraction:

• Multiply 2 by 7 to get 14 (\( \frac{2 \times 7}{5 \times 7} = \frac{14}{35} \)).

The fraction \( \frac{2}{35} \) already has 35 as its denominator, so it remains unchanged. Converting fractions in this way aligns all of them for easy addition or subtraction.

Subtraction and Addition of Fractions

With all fractions having the same denominator, it's straightforward to perform subtraction and addition. Simply keep the denominator the same and operate on the numerators.
In this problem, start by subtracting the transformed fractions: \( \frac{15}{35} - \frac{14}{35} \). Subtract the numerators, 15 and 14, resulting in \( \frac{1}{35} \).
Next, add \( \frac{2}{35} \) to \( \frac{1}{35} \). This means adding the numerators once again: 1 + 2 equals 3. Thus, \( \frac{1}{35} + \frac{2}{35} = \frac{3}{35} \).
Completing the operation, always check whether the final fraction can be simplified. Here, since 3 and 35 have no common factors other than 1, \( \frac{3}{35} \) remains as is. This method illustrates how to handle multiple fractions efficiently using common denominators.