Problem 11 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{x}{3}+\frac{2 x}{7} $$

Step-by-Step Solution

Answer

\(\frac{13x}{21}\) is the simplified form.

Step 1: Identify the Least Common Denominator (LCD)

The denominators in the expression are 3 and 7. The least common denominator of these numbers is 21.

Step 2: Convert Each Fraction to Have the LCD

To make the denominators the same, convert each fraction by multiplying the numerator and the denominator by the necessary value. For \(\frac{x}{3}\), multiply by \(\frac{7}{7}\) to get \(\frac{7x}{21}\).For \(\frac{2x}{7}\), multiply by \(\frac{3}{3}\) to get \(\frac{6x}{21}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them together:\[\frac{7x}{21} + \frac{6x}{21} = \frac{(7x + 6x)}{21} = \frac{13x}{21}\]

Step 4: Simplify the Fraction

In this case, \(\frac{13x}{21}\) is already in its simplest form because 13 and 21 have no common factors other than 1.

Key Concepts

Least Common Denominator

When working with fractions that have different denominators, the first step is finding the Least Common Denominator (LCD). The LCD is the smallest number that each denominator can divide without leaving a remainder.
For fractions like \( \frac{x}{3} \) and \( \frac{2x}{7} \), the denominators are 3 and 7. To find their LCD, we list some multiples of each number:

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
• Multiples of 7: 7, 14, 21, 28, ...

The smallest common multiple in both lists is 21. Thus, 21 is the least common denominator for these fractions.
Finding the LCD is vital because it allows us to convert fractions into like terms, making them easier to add or subtract. The next step involves transforming each fraction so that they share this common denominator, which is necessary for proceeding with operations such as addition.

Fraction Addition

Once the least common denominator is identified, the next step is to perform fraction addition. To add fractions, they must have the same denominator. Here's how we do it.
First, we adjust each fraction to have the LCD as the new denominator. For \( \frac{x}{3} \), multiply both the numerator and the denominator by 7, giving \( \frac{7x}{21} \). Similarly, for \( \frac{2x}{7} \), multiply by 3 to get \( \frac{6x}{21} \). Both fractions now have 21 as the denominator.
This process involves converting each fraction by multiplying by 1 in a clever form, like \( \frac{7}{7} \) or \( \frac{3}{3} \). This step is crucial because it allows us to perform the addition by making the denominators equal.
Finally, with the denominators aligned, simply add the numerators: \( \frac{7x}{21} + \frac{6x}{21} = \frac{13x}{21} \). The calculation becomes straightforward as we now only focus on the numerators while the denominators remain the same.

Simplification of Fractions

After adding fractions, the last key step is simplifying the result. Simplifying consists of reducing a fraction to its simplest form, where possible.
To simplify a fraction, check if the numerator and the denominator have any common factors besides 1. If they do, divide both by their greatest common factor.
In our example, the fraction \( \frac{13x}{21} \) is already in its simplest form. This is because 13 and 21 share no common factors other than 1. To verify, factorize both numbers:

• 13 is a prime number, so its only factors are 1 and 13.
• 21 factors into 1, 3, 7, and 21.

Seeing no common factors means further simplification isn't possible here.
Thus, always check for simplification to ensure your result is as easy to interpret and concise as possible.