Problem 47 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Write the answer in lowers ferms. See Example 7. $$ \frac{2}{3}+\frac{3}{7} $$

Step-by-Step Solution

Answer

The sum is \(\frac{23}{21}\) (an improper fraction).

Step 1: Find a Common Denominator

To add the fractions \(\frac{2}{3}\) and \(\frac{3}{7}\), we first need to find a common denominator. The denominators 3 and 7 are coprime, so the least common multiple is their product, which is 21. Thus, the common denominator is 21.

Step 2: Rewrite Fractions with Common Denominator

Convert each fraction to an equivalent fraction with the common denominator of 21. - For \(\frac{2}{3}\), multiply both the numerator and denominator by 7: \[\frac{2 \times 7}{3 \times 7} = \frac{14}{21}\]- For \(\frac{3}{7}\), multiply both the numerator and denominator by 3:\[\frac{3 \times 3}{7 \times 3} = \frac{9}{21}\]

Step 3: Add the Fractions

With both fractions expressed with a common denominator, simply add the numerators and keep the denominator the same:\[\frac{14}{21} + \frac{9}{21} = \frac{23}{21}\]

Step 4: Check for Simplest Form

The result \(\frac{23}{21}\) is already in simplest form because the numerator 23 and the denominator 21 have no common factors other than 1.

Step 5: Conclusion

The sum of \(\frac{2}{3}\) and \(\frac{3}{7}\) is \(\frac{23}{21}\), which is an improper fraction.

Key Concepts

Understanding the Least Common Denominator

When adding or subtracting fractions, it's essential to work with a common denominator. The common denominator allows us to combine fractions as they represent parts of a whole in a compatible way. To find the least common denominator, look for the smallest number that is a multiple of both denominators involved.

• For instance, with fractions \(\frac{2}{3}\) and \(\frac{3}{7}\), the denominators are 3 and 7.
• These numbers are coprime, meaning they have no common factors except for 1.
• Thus, their least common denominator is simply their product, 21.

Finding this common denominator is a pivotal first step in ensuring the fractions can be easily combined.

Improper Fractions Explained

An improper fraction occurs when the numerator (top number) is greater than or equal to the denominator (bottom number). This often happens after performing operations like addition on fractions.
In our example, after adding the fractions \(\frac{14}{21} + \frac{9}{21}\), we obtain \(\frac{23}{21}\) as the result.

• Here, the numerator 23 is larger than the denominator 21, making it an improper fraction.
• Improper fractions can sometimes be converted to mixed numbers for easier interpretation, but this isn’t always necessary, depending on the context.
• In simplification exercises, it’s crucial to recognize when a fraction is improper.

Simplifying Fractions

Simplifying fractions, or reducing them to their simplest form, involves making the numerator and denominator as small as possible while maintaining the same value of the fraction.

• This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• In the case of \(\frac{23}{21}\), the GCD of 23 and 21 is 1, indicating that the fraction is already in simplest form.

Simplification ensures that the fraction is represented in the most basic terms possible, making calculations and interpretations more straightforward.

Equivalent Fractions

Equivalent fractions are different fractions that express the same value or proportion. Understanding this concept is fundamental when finding a common denominator.

• For example, \(\frac{2}{3}\) can be rewritten as \(\frac{14}{21}\) by multiplying the numerator and denominator by 7.
• Similarly, \(\frac{3}{7}\) is equivalent to \(\frac{9}{21}\) when both parts of the fraction are multiplied by 3.
• Each of these operations maintains the original value of the fraction even though the numbers look different.

Grasping equivalent fractions is key to performing and understanding operations like addition and subtraction with fractions.