Problem 66 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{7}{15}-\frac{7}{25} $$

Step-by-Step Solution

Answer

The result of \( \frac{7}{15} - \frac{7}{25} \) is \( \frac{14}{75} \).

Step 1: Identify the problem

We need to subtract two fractions: \( \frac{7}{15} - \frac{7}{25} \). To perform this operation, the fractions must have the same denominator.

Step 2: Find a common denominator

The denominators of the fractions are 15 and 25. To find the least common denominator (LCD), we calculate the least common multiple (LCM) of 15 and 25. The LCM of 15 and 25 is 75.

Step 3: Convert fractions to have the common denominator

Change each fraction to have the common denominator of 75. For \( \frac{7}{15} \), multiply the numerator and the denominator by 5: \( \frac{7 \times 5}{15 \times 5} = \frac{35}{75} \). For \( \frac{7}{25} \), multiply the numerator and the denominator by 3: \( \frac{7 \times 3}{25 \times 3} = \frac{21}{75} \).

Step 4: Perform the subtraction

Subtract the numerators of the converted fractions: \( \frac{35}{75} - \frac{21}{75} = \frac{35 - 21}{75} = \frac{14}{75} \).

Step 5: Simplify the result (if necessary)

The fraction \( \frac{14}{75} \) is already in its simplest form, as 14 and 75 have no common factors besides 1.

Key Concepts

least common denominator

When subtracting fractions, it is important to work with common denominators. This is because fractions can only be added or subtracted when they share the same denominator, which represents equal parts of a whole. The least common denominator (LCD) is the smallest common multiple of the denominators of the fractions you are working with.

• For fractions like \( \frac{7}{15} \) and \( \frac{7}{25} \), we first identify the denominators: 15 and 25.
• We then find a number that both 15 and 25 can divide evenly, which is known as the least common multiple (LCM).
• The LCM of 15 and 25 is 75, making it our LCD.

By converting each fraction to equivalent fractions with this LCD, you ensure that the fractions are accurate and comparable. This step lays the groundwork for seamless addition or subtraction.

subtracting fractions

Once you have transformed fractions to have the same least common denominator, subtracting them becomes straightforward. This operation involves subtracting only the numerators, as the denominator is now common and remains unchanged.

• Consider the fractions \( \frac{35}{75} \) and \( \frac{21}{75} \), both sharing the denominator 75.
• Subtract the second numerator from the first: \( 35 - 21 = 14 \).
• Keep the common denominator: \( \frac{14}{75} \).

With a consistent denominator across all fractions, the focus is solely on the numerators, simplifying the task considerably and ensuring the result is accurate.

simplifying fractions

Simplifying fractions is the process of finding an equivalent fraction with the smallest numbers possible, known as the simplest form. A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

• Take the fraction \( \frac{14}{75} \), the result from our subtraction operation.
• Check for common factors; here, 14 and 75 have no common factors—except 1—indicating \( \frac{14}{75} \) is already simplified.
• If common factors existed, you would divide both the numerator and denominator by the greatest common factor to simplify.

Simplifying fractions makes it easier to understand and work with them, ensuring your answer is as clear and precise as possible.