Problem 63 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{10}{3}-\frac{5}{21} $$

Step-by-Step Solution

Answer

The result of \( \frac{10}{3} - \frac{5}{21} \) is \( \frac{65}{21} \).

Step 1: Find the Least Common Denominator

The denominators of the fractions are 3 and 21. The least common multiple of 3 and 21 is 21. Therefore, the least common denominator for these fractions is 21.

Step 2: Convert to Common Denominator

Rewrite each fraction with the least common denominator of 21. For \( \frac{10}{3} \), multiply both the numerator and the denominator by 7 to get \( \frac{70}{21} \). The fraction \( \frac{5}{21} \) is already over the denominator 21.

Step 3: Perform the Subtraction

Now subtract the fractions using the common denominator: \( \frac{70}{21} - \frac{5}{21} = \frac{70 - 5}{21} = \frac{65}{21} \).

Step 4: Simplify the Fraction

The fraction \( \frac{65}{21} \) needs to be simplified. Both 65 and 21 have no common factors other than 1, so \( \frac{65}{21} \) is already in its simplest form.

Key Concepts

Least Common Denominator

When subtracting or adding fractions, denominators must match to perform the operation seamlessly. This is where the concept of the Least Common Denominator (LCD) comes into play. The LCD is essentially the smallest common multiple of the fractions' denominators. In the problem \( \frac{10}{3} - \frac{5}{21} \), the denominators are 3 and 21.
To find the LCD:

• List the multiples of each denominator.
• 3: 3, 6, 9, 12, 15, 18, 21...
• 21: 21, 42, 63...

Looking at these, the smallest or least multiple that both lists share is 21. Thus, the LCD is 21. Finding the LCD allows us to rewrite each fraction with this common denominator, which is crucial for easy subtraction or addition.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. A fraction is in simplest form when the numerator and the denominator have no common factors other than 1. In the problem provided, after performing the subtraction, the result was \( \frac{65}{21} \).
To determine if \( \frac{65}{21} \) can be simplified:

• Check the greatest common divisor (GCD) of 65 and 21.
• List factors of 65: 1, 5, 13, 65.
• List factors of 21: 1, 3, 7, 21.
• The GCD is 1 since no other number divides both 65 and 21.

Since the only mutual factor is 1, \( \frac{65}{21} \) is already simplified. Always simplify your final answer if possible, which helps in making the fraction easier to understand and use.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value or proportion of the whole. To create equivalent fractions, both the numerator and the denominator of a fraction can be multiplied by the same non-zero integer. In our example, to transform \( \frac{10}{3} \) to have the common denominator 21:

• Multiply both the numerator and the denominator by 7 to yield \( \frac{70}{21} \).
• \( \frac{5}{21} \) doesn't need any changes since it's already in terms of the common denominator.

Creating equivalent fractions is essential when adding or subtracting fractions with different denominators. It ensures each part of the fraction represents the same total, allowing us to perform mathematical operations smoothly. Always check the conversion by simplifying back and ensuring the original and equivalent fractions are indeed equal.