Problem 90 - Elementary Algebra

Question

To add \(\frac{5}{6} x\) and \(\frac{1}{4} x\), the fractions can be rewritten with the common denominator 24 , or they can be rewritten with the least common denominator, 12. Explain the advantage of using the least common denominator.

Step-by-Step Solution

Answer

The least common denominator simplifies the calculation by using smaller numbers for easier arithmetic.

Step 1 - Identify the Fractions and Variables

The given fractions are \(\frac{5}{6} x\) and \(\frac{1}{4} x\). Each fraction has a variable, \(x\), which is common.

Step 2 - Determine the Least Common Denominator (LCD)

To add fractions, find the least common denominator (LCD). The denominators here are 6 and 4. Factorize them: \(6 = 2 \times 3\) and \(4 = 2 \times 2\). The least common denominator is the smallest number that is a multiple of both denominators, which is 12.

Step 3 - Rewrite Fractions with the Least Common Denominator

Convert each fraction to have the least common denominator (12):\[\frac{5}{6} x = \frac{5 \times 2}{6 \times 2} x = \frac{10}{12} x\]\[\frac{1}{4} x = \frac{1 \times 3}{4 \times 3} x = \frac{3}{12} x\]

Step 4 - Add the Fractions

Now that both fractions have the same denominator, add them together: \(\frac{10}{12} x + \frac{3}{12} x = \frac{13}{12} x\)

Step 5 - Explain the Advantage of Using the LCD

Using the least common denominator (12) simplifies the calculation. It results in smaller numbers, making the addition easier and reducing the steps required.

Key Concepts

least common denominator

When adding fractions like \(\frac{5}{6} x\) and \(\frac{1}{4} x\), finding a common denominator is crucial. A common denominator is simply a shared multiple of the denominators. The least common denominator (LCD) specifically is the smallest common multiple. For the fractions above, the denominators are 6 and 4.
Factorize these numbers: \[6 = 2 \times 3\] and \[4 = 2 \times 2\].
The smallest number that is a multiple of both 6 and 4 is 12. Therefore, the LCD is 12. Using the LCD makes calculations easier by reducing the size of numbers involved, thereby simplifying the arithmetic operations needed. This allows us to add fractions with smaller, more manageable numbers.

fraction addition

Now, let's understand fraction addition. Imagine you want to add \(\frac{5}{6} x\) and \(\frac{1}{4} x\). Once the fractions have a common denominator, you can easily add them:
Convert each fraction to have the least common denominator (12):\[ \frac{5}{6} x = \frac{5 \times 2}{6 \times 2} x = \frac{10}{12} x \] and\[ \frac{1}{4} x = \frac{1 \times 3}{4 \times 3} x = \frac{3}{12} x \].
Next, add the numerators while keeping the denominator the same: \[ \frac{10}{12} x + \frac{3}{12} x = \frac{13}{12} x \].
This means both fractions add up to \(\frac{13}{12} x\).
See how straightforward the addition becomes once the denominators are the same?

simplifying fractions

After adding fractions, it's beneficial to simplify the result if possible. Simplifying a fraction means rewriting it in its simplest form.
In our example, you might have a fraction like \(\frac{13}{12} x\). This fraction is already simplified because 13 is a prime number and does not share any common factors with 12 other than 1.

However, if you had a fraction like \(\frac{10}{12} x\), you could simplify it further. Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2: \[ \frac{10}{12} x = \frac{10 \div 2}{12 \div 2} x = \frac{5}{6} x \].
Simplifying helps in making the fraction more understandable and can reveal any deeper relationships in the numbers involved.