Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \frac{5}{7}-\frac{1}{13} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{58}{91}\).

Step 1: Find the Least Common Denominator (LCD)

To subtract these fractions, we need a common denominator. The denominators are 7 and 13. The least common denominator of 7 and 13 is their product since they are both prime numbers: \[ 7 \times 13 = 91 \]

Step 2: Convert Fractions to Equivalent Fractions with the LCD

Now, we'll rewrite both fractions with the denominator of 91:\[ \frac{5}{7} = \frac{5 \times 13}{7 \times 13} = \frac{65}{91} \]\[ \frac{1}{13} = \frac{1 \times 7}{13 \times 7} = \frac{7}{91} \]

Step 3: Subtract the Equivalent Fractions

With the same denominators, subtract the numerators directly:\[ \frac{65}{91} - \frac{7}{91} = \frac{65 - 7}{91} = \frac{58}{91} \]

Step 4: Simplify the Fraction

Check if \(\frac{58}{91}\) can be simplified. Since 58 and 91 have no common divisors other than 1, it is already in its simplest form.

Key Concepts

Least Common Denominator (LCD)

The Least Common Denominator, or LCD, is crucial when performing operations like addition and subtraction on fractions. It ensures that we're working with equivalent pieces across all fractions involved. The LCD for two or more fractions is the smallest number that is a multiple of each of their denominators.

In our example, we have two denominators: 7 and 13. Both of these are prime numbers, meaning they don't have any divisors other than 1 and themselves. Therefore, the simplest way to find the least common denominator is by multiplying the denominators together. So, the LCD is \[ 7 \times 13 = 91 \]
Remember:

• The LCD is essential for combining fractions accurately.
• When dealing with prime denominators, multiplying them gives the LCD.

Working with the LCD lets us transform each fraction into an equivalent form that can be easily subtracted or added.

Subtraction of Fractions

Subtracting fractions becomes straightforward once they share a common denominator. The process is quite simple as you only need to subtract the numerators while the denominator remains unchanged. This is due to the concept of equivalent parts that the LCD creates for us.

Let's walk through the exercise:

• First, find the equivalent form of each fraction using the LCD. Here, we transform \( \frac{5}{7} \) and \( \frac{1}{13} \) to \( \frac{65}{91} \) and \( \frac{7}{91} \), respectively.
• Now, subtract the numerators: \[ 65 - 7 = 58 \]
• Keep the denominator the same: \[ \frac{58}{91} \]

The final fraction, \( \frac{58}{91} \), is the result of the subtraction of these equivalent fractions. This approach guarantees the result is accurate and simplifies the potentially confusing operation.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. They play a pivotal role in operations like addition or subtraction by aligning denominators. Finding equivalent fractions means transforming a given fraction so it fits the common denominator (in this case, the LCD).

For our exercise, we create equivalent fractions as follows:

• For \( \frac{5}{7} \), align it with the denominator 91: \[ \frac{5}{7} = \frac{5 \times 13}{7 \times 13} = \frac{65}{91} \]
• For \( \frac{1}{13} \), do likewise: \[ \frac{1}{13} = \frac{1 \times 7}{13 \times 7} = \frac{7}{91} \]

It's important to understand:

• Equivalent fractions allow for the subtraction of fractions since they standardize the size of the pieces involved.
• Transforming fractions while maintaining their value is key to solving fraction-related problems accurately.

Once you practice creating equivalent fractions, it becomes second nature, boosting your confidence in fraction operations.