Problem 125 - Intermediate Algebra

Question

Simplify: $$\frac{x}{2}+\frac{x}{3}+\frac{x}{4}+\frac{x}{5}+\frac{x}{6}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{29x}{20} \).

Step 1: Combine Terms with Common Denominator

Let's first find a common denominator for all the terms. The denominators are 2, 3, 4, 5, and 6. The smallest common multiple of these numbers is 60. So, rewrite each term with a denominator of 60.

Step 2: Rewrite Each Term with Denominator of 60

Convert each term so they have the denominator of 60:\[ \frac{x}{2} = \frac{30x}{60}, \quad \frac{x}{3} = \frac{20x}{60}, \quad \frac{x}{4} = \frac{15x}{60}, \quad \frac{x}{5} = \frac{12x}{60}, \quad \text{and} \quad \frac{x}{6} = \frac{10x}{60} \]

Step 3: Add the Fractions

Now, add up all the terms:\[ \frac{30x}{60} + \frac{20x}{60} + \frac{15x}{60} + \frac{12x}{60} + \frac{10x}{60} \]Combine the numerators:\[ \frac{30x + 20x + 15x + 12x + 10x}{60} = \frac{87x}{60} \]

Step 4: Simplify the Fraction

Finally, simplify \( \frac{87x}{60} \) by finding the greatest common divisor of 87 and 60, which is 3:\[ \frac{87x}{60} = \frac{29x}{20} \]The simplified expression is \( \frac{29x}{20} \).

Key Concepts

Least Common Multiple

When dealing with fractions that have different denominators, finding the least common multiple (LCM) helps us combine them. The LCM of a set of numbers is the smallest number that each of the numbers can divide evenly into. For example, if we have denominators like 2, 3, 4, 5, and 6 as in the original problem, we look for the smallest number that all these can divide evenly.

• 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, ...
• 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, ...
• 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, ...
• 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...
• 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...

As we can see, 60 is the smallest number present in each list, making it the LCM. Using the LCM as a common denominator enables us to combine fractions into a single fraction, thus simplifying calculation.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can evenly divide each of two numbers. In fraction simplification, we use the GCD to simplify the numerator and denominator of a fraction to its simplest form. Let's take the fraction \( \frac{87x}{60} \) from our example. To simplify it, we first need to find the GCD of 87 and 60.

• Factors of 87: 1, 3, 29, 87
• Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Here, the largest common factor is 3. Divide both the numerator and denominator by 3:\[\frac{87x}{60} = \frac{87x \div 3}{60 \div 3} = \frac{29x}{20}\]This gives us the fraction in its simplest form, making it more manageable and easier to understand.

Common Denominator

To add or subtract fractions, a common denominator is needed. This is because fractions must have the same base (denominator) when adding or subtracting. Without a common denominator, combining fractions involves complex calculations, potentially leading to errors.In the original problem, we combined fractions by finding the least common multiple of the denominators. By rewriting each fraction with 60 as a common denominator:

• \( \frac{x}{2} = \frac{30x}{60} \)
• \( \frac{x}{3} = \frac{20x}{60} \)
• \( \frac{x}{4} = \frac{15x}{60} \)
• \( \frac{x}{5} = \frac{12x}{60} \)
• \( \frac{x}{6} = \frac{10x}{60} \)

we simplified the problem by converting all fractions to a common base. At this point, adding the fractions becomes a straightforward task involving only the numerators.

Combining Fractions

Combining fractions involves adding or subtracting fractions that have a common denominator. Once a common denominator is established, combining fractions focuses on summing or subtracting their numerators.In the context of the given problem, here’s how we combine fractions:

• Add numerators: \(30x + 20x + 15x + 12x + 10x = 87x\)
• Keep the common denominator: 60
• Combine as a single fraction: \(\frac{87x}{60}\)

The result, \(\frac{87x}{60}\), is a single fraction representing the sum of the originally separate fractions. By using a common denominator, we streamline the process to achieve a simplified solution. Once combined, simplify the result whenever possible to make the calculation more efficient.